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Applications of second order differential equations

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  2. What are some applications of second order differential equations?
  3. Mechanical Vibrations and Dampening Forces
  4. Damping Forces on Mechanical Vibrations
  5. Electrical Circuits
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Examples
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  1. Mechanical Vibrations

    A spring has a weight of 5kg attached to the end of it. The spring has a natural length of 0.3m and a force of 35 newtons is required to stretch the spring to a length 0.8m. If the spring is stretched to 0.5 meters and then released (with zero initial velocity), then what is the position of the mass at time tt?
    1. Suppose that a hydraulic shock has a spring constant of 40 newtons per meter. There is a weight of 10kg attached to the end of the shock, and the shock has a resting length of 0.5 meters.
      1. What is the position of the mass at time tt if the hydraulic shock has a damping constant of c=50c=50, with an initial positions of 0.75 meters, and an initial velocity of 0 m/sm/s?
      2. What is the position of the mass at time tt if the hydraulic shock has a damping constant of c=40c=40, with an initial position of 0.5 meters, and an initial velocity of 5 m/sm/s?
      3. What is the position of the mass at time tt if the hydraulic shock has a damping constant of c=20c=20, with an initial position of 0.3 meters, and an initial velocity of -0.3 m/sm/s?

    2. Electrical Circuits

      Find the charge at time tt for an electrical circuit with a resistor that has a resistance of R=14ΩR=14 \Omega , an inductor with L=2HL=2H, a capacitor with C=0.05FC=0.05 F, and a battery with charge E(t)=8E(t)=8sin⁡(2t)\sin(2t). The initial charge is V=2229V=\frac{22}{29} coulombs, and the initial current is I=629I=\frac{6}{29} amps.

      Topic Notes
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      Introduction to Applications of Second Order Differential Equations

      Second order differential equations play a crucial role in modeling various real-world phenomena, particularly in the fields of mechanical vibrations and electrical circuits. These equations provide powerful tools for analyzing and predicting the behavior of complex systems. In mechanical vibrations, second order differential equations are used to describe the motion of objects under the influence of forces, such as springs and dampers. They help engineers design everything from suspension systems in vehicles to earthquake-resistant structures. In electrical circuits, these equations model the flow of current and voltage in systems with inductors and capacitors, enabling the design of filters, oscillators, and other essential components. The introduction video accompanying this section offers a visual and intuitive understanding of these applications, demonstrating how second order differential equations bridge the gap between mathematical theory and practical engineering problems. By mastering these concepts, students gain valuable insights into the fundamental principles governing many physical systems, preparing them for advanced studies in engineering and applied mathematics.

      Mechanical Vibrations: Basic Concepts and Equations

      Mechanical vibrations are a fundamental concept in engineering and physics, describing the oscillatory motion of objects or systems. At the heart of understanding these vibrations lies the application of second order differential equations, which provide a mathematical framework for analyzing and predicting the behavior of vibrating systems. Two key principles, Hooke's Law and Newton's Second Law, form the foundation for deriving these equations and understanding the dynamics of spring-mass systems.

      Hooke's Law, named after the 17th-century physicist Robert Hooke, states that the force exerted by a spring is directly proportional to its displacement from equilibrium. Mathematically, this is expressed as F = -kx, where F is the restoring force, k is the spring constant (a measure of the spring's stiffness), and x is the displacement. The negative sign indicates that the force acts in the opposite direction of the displacement, always trying to return the system to its equilibrium position.

      Newton's Second Law, a cornerstone of classical mechanics, relates force to mass and acceleration. It states that the net force acting on an object is equal to the object's mass multiplied by its acceleration: F = ma, where F is the net force, m is the mass, and a is the acceleration. This law is crucial in describing the motion of objects under the influence of forces, including those in vibrating systems.

      When we combine Hooke's Law and Newton's Second Law, we can derive the basic equation for mechanical vibrations in a spring-mass system. Consider a mass m attached to a spring with spring constant k. As the mass oscillates, it experiences a restoring force from the spring (given by Hooke's Law) and potentially other forces like damping or external excitations. Applying Newton's Second Law to this system, we can write:

      m(d²x/dt²) = -kx - c(dx/dt) + F(t)

      Here, d²x/dt² represents the acceleration of the mass, -kx is the spring force, -c(dx/dt) is a damping force (where c is the damping coefficient), and F(t) represents any external force applied to the system. This equation is a second order differential equation, as it involves the second derivative of position with respect to time.

      For a simple undamped spring-mass system without external forces, the equation simplifies to:

      m(d²x/dt²) + kx = 0

      This is the fundamental equation for simple harmonic motion, a special case of mechanical vibrations where the restoring force is directly proportional to displacement. The solution to this equation describes the position of the mass as a function of time, typically in the form of sinusoidal functions.

      The natural frequency of the system, denoted as ω_n, is given by ω_n = (k/m). This frequency represents how quickly the system oscillates when disturbed from its equilibrium position and then left to vibrate freely. The period of oscillation, T, is related to the natural frequency by T = 2π/ω_n.

      Understanding these basic concepts and equations is crucial for analyzing more complex vibrating systems. In real-world applications, engineers often deal with multiple degrees of freedom, non-linear systems, and forced vibrations. However, the principles of Hooke's Law and Newton's Second Law, along with the derived second order differential equations, remain at the core of mechanical vibration analysis.

      The study of mechanical vibrations has wide-ranging applications in engineering and technology. It's essential in designing structures that can withstand dynamic loads, developing vibration isolation systems, and analyzing the behavior of mechanical and structural systems under various operating conditions. From the design of earthquake-resistant buildings to the development of precision machinery and musical instruments, the principles of mechanical vibrations play a crucial role.

      In more advanced studies, concepts like modal analysis, Fourier transforms, and vibration control techniques build upon these fundamental principles. These advanced topics allow for the analysis of complex, multi-degree-of-freedom systems and the development of sophisticated vibration control strategies. However, a solid understanding of the basic spring-mass system and its governing differential equation remains the foundation for all these advanced applications.

      In conclusion, the

      Damped Mechanical Vibrations

      Damping is a crucial concept in mechanical vibrations, playing a significant role in controlling and modifying the motion of oscillating systems. In the context of a spring-mass system, damping refers to the dissipation of energy that occurs during oscillation, gradually reducing the amplitude of vibration over time. This phenomenon is essential in various engineering applications, from shock absorbers in vehicles to seismic protection in buildings.

      To understand damping, we must first consider an undamped spring-mass system. In this idealized case, a mass attached to a spring oscillates indefinitely with constant amplitude, governed by Hooke's law. The equation of motion for an undamped system is:

      m(d²x/dt²) + kx = 0

      Where m is the mass, k is the spring constant, and x is the displacement. This equation results in simple harmonic motion with a natural frequency of ω = (k/m).

      However, real-world systems experience damping forces that oppose motion. The most common form is viscous damping, where the damping force is proportional to the velocity of the mass. When we introduce damping to our spring-mass system, the equation of motion becomes:

      m(d²x/dt²) + c(dx/dt) + kx = 0

      Here, c is the damping coefficient, and the term c(dx/dt) represents the damping force. This equation describes damped mechanical vibrations, where the system's behavior depends on the relative magnitudes of m, c, and k.

      The solution to this differential equation takes the form:

      x(t) = Ae^(-ζωt) cos(ω_d t + φ)

      Where ζ is the damping ratio (ζ = c / (2(km))), ω_d is the damped natural frequency (ω_d = ω(1 - ζ²)), A is the initial amplitude, and φ is the phase angle. The damping ratio ζ determines the system's behavior:

      • If ζ < 1: Underdamped system (oscillatory decay)
      • If ζ = 1: Critically damped system (fastest return to equilibrium without oscillation)
      • If ζ > 1: Overdamped system (non-oscillatory decay)

      Comparing damped and undamped systems reveals several key differences:

      1. Amplitude: In undamped systems, the amplitude remains constant. In damped systems, it decreases exponentially over time.
      2. Frequency: Undamped systems oscillate at their natural frequency ω. Damped systems have a slightly lower frequency ω_d.
      3. Energy: Undamped systems conserve energy, while damped systems dissipate energy over time.
      4. Long-term behavior: Undamped systems continue to oscillate indefinitely, while damped systems eventually come to rest at the equilibrium position.

      The damping force significantly affects the motion of a spring-mass system by introducing energy dissipation. This results in a gradual decrease in the amplitude of oscillations, with the rate of decay determined by the damping coefficient. The damping force also slightly reduces the frequency of oscillation compared to the undamped case.

      In practical applications, engineers often aim to achieve optimal damping for specific purposes. For instance, in vehicle suspension systems, underdamping can lead to a bouncy ride, while overdamping may result in a stiff, uncomfortable experience. Critical damping is often desirable in scenarios where rapid stabilization is crucial, such as in precision instruments or robotic arms.

      Understanding damped mechanical vibrations is essential for designing and analyzing various mechanical and structural systems. By carefully selecting the damping coefficient in relation to the mass and spring constant, engineers can control the system's response to external forces, improve stability, and enhance overall performance. This knowledge is applied in

      Electrical Circuits: Introduction and Basic Components

      Second order differential equations play a crucial role in the analysis and understanding of electrical circuits, particularly in the study of RLC circuits. These equations provide a mathematical framework to describe the behavior of electrical components and their interactions within a circuit. In this section, we'll explore the application of these equations to electrical circuits, introduce the basic components of an RLC circuit, and discuss the importance of Kirchhoff's Law in circuit analysis.

      An RLC circuit is a fundamental electrical circuit consisting of three key components: a resistor (R), an inductor (L), and a capacitor (C). Each of these components has a unique role in shaping the circuit's behavior:

      1. Resistor (R): This component opposes the flow of electric current in a circuit. It converts electrical energy into heat through a process called Joule heating. The resistance is measured in ohms (Ω) and is governed by Ohm's Law, which states that the voltage across a resistor is directly proportional to the current flowing through it.

      2. Inductor (L): An inductor stores energy in a magnetic field when current flows through it. It opposes changes in current and is characterized by its inductance, measured in henries (H). The voltage across an inductor is proportional to the rate of change of current through it, described by the equation V = L(dI/dt).

      3. Capacitor (C): This component stores electric charge and energy in an electric field. It is characterized by its capacitance, measured in farads (F). The voltage across a capacitor is proportional to the charge stored on its plates, given by the equation V = Q/C, where Q is the charge and C is the capacitance.

      The interaction of these components in an RLC circuit gives rise to complex behavior that can be described using second order differential equations. These equations take into account the relationships between voltage, current, and the properties of each component, allowing engineers to predict and analyze circuit behavior over time.

      One of the most fundamental principles in circuit analysis is Kirchhoff's Law, which consists of two key rules:

      1. Kirchhoff's Current Law (KCL): This law states that the sum of currents entering a node in a circuit must equal the sum of currents leaving that node. In other words, the algebraic sum of currents at any junction in a circuit is zero.

      2. Kirchhoff's Voltage Law (KVL): This law states that the sum of all voltages around any closed loop in a circuit must equal zero. This principle is based on the energy conservation in circuits and ensures that the total energy in a circuit is conserved.

      Kirchhoff's Law is of paramount importance in circuit analysis for several reasons:

      1. It provides a systematic approach to analyzing complex circuits by breaking them down into simpler loops and nodes.

      2. It allows for the formulation of equations that describe the behavior of currents and voltages throughout the circuit.

      3. It forms the basis for many circuit analysis techniques, such as mesh analysis and nodal analysis.

      4. It helps in understanding the distribution of energy conservation in circuits and ensures that the principles of energy conservation are maintained.

      When applying second order differential equations to RLC circuits, Kirchhoff's Law is used to set up the initial equations that describe the circuit's behavior. These equations typically involve the voltages across each component and the currents flowing through them. By combining these equations with the characteristic equations of each component (such as Ohm's Law for resistors), a second order differential equation can be derived that describes the overall circuit behavior.

      For example, in a series RLC circuit, the second order differential equation for the charge q(t) on the capacitor can be written as:

      L(d²q/dt²) + R(dq/dt) + (1/C)q = V(t)

      Where V(t) is the applied voltage source. This equation encapsulates the interplay between

      Deriving the Second Order Differential Equation for RLC Circuits

      The process of deriving the second order differential equation for an RLC circuit is a fundamental concept in electrical engineering, combining principles of circuit analysis and mathematical modeling. An RLC circuit consists of three key components: a resistor (R), an inductor (L), and a capacitor (C). To understand the behavior of this circuit, we need to analyze the voltage drops across each component and how they contribute to the overall circuit dynamics.

      Let's begin by examining the voltage drops across each component. For the resistor, we apply Ohm's Law, which states that the voltage drop (VR) is proportional to the current (I) flowing through it: VR = IR. The inductor's voltage drop (VL) is governed by Faraday's Law of Induction, which relates the voltage to the rate of change of current: VL = L(dI/dt). Finally, the capacitor's voltage (VC) is related to the charge (Q) stored on its plates: VC = Q/C.

      To derive the second order differential equation, we apply Kirchhoff's Voltage Law (KVL) to the RLC circuit. KVL states that the sum of voltage drops around a closed loop must equal zero. In our RLC circuit, we can express this as: VR + VL + VC = Vsource, where Vsource is the applied voltage.

      Substituting the expressions for each voltage drop, we get: IR + L(dI/dt) + Q/C = Vsource. To transform this into a second order differential equation, we need to express everything in terms of charge Q. We know that current I is the rate of change of charge, so I = dQ/dt. Substituting this and taking the derivative of both sides with respect to time, we obtain: R(dQ/dt) + L(d²Q/dt²) + Q/C = dVsource/dt.

      Rearranging the terms, we arrive at the second order differential equation for the RLC circuit: L(d²Q/dt²) + R(dQ/dt) + Q/C = dVsource/dt. This equation describes the behavior of charge in the circuit over time, taking into account the effects of resistance, inductance, and capacitance.

      Interestingly, this equation bears a striking resemblance to the equation describing mechanical vibrations in a spring-mass-damper system. In the mechanical analogy, mass corresponds to inductance, damping to resistance, and spring stiffness to the inverse of capacitance. The equation for mechanical vibrations takes the form: m(d²x/dt²) + b(dx/dt) + kx = F(t), where m is mass, b is damping coefficient, k is spring constant, x is displacement, and F(t) is the applied force.

      The similarity between these equations highlights the fundamental connections between electrical and mechanical systems. Both describe second order systems that can exhibit oscillatory behavior, damping, and resonance. This analogy allows engineers and physicists to apply insights from one domain to the other, facilitating a deeper understanding of both electrical and mechanical systems.

      In practical applications, the solving RLC circuit equation is crucial for analyzing and designing various electrical systems. It helps in understanding the transient response of circuits, determining resonant frequencies, and optimizing circuit performance. Engineers use this equation to design filters, oscillators, and power supplies, among other applications.

      To solve the RLC circuit equation, various methods can be employed, including Laplace transforms, phasor analysis for steady-state AC circuits, and numerical methods for more complex scenarios. The solution provides information about the circuit's behavior, such as its natural frequency, damping factor, and response to different input signals.

      In conclusion, deriving the second order differential equation for an RLC circuit involves analyzing the voltage drops across each component and applying fundamental laws of electrical circuits. The resulting equation not only describes

      Solving Second Order Differential Equations for Mechanical and Electrical Systems

      Second order differential equations play a crucial role in modeling and analyzing mechanical vibrations and electrical circuits. Understanding the methods for solving these equations is essential for engineers and physicists working with dynamic systems. This section will explore various techniques for solving second order differential equations and demonstrate their applications in real-world scenarios.

      One of the most common methods for solving second order differential equations is the characteristic equation method. This approach involves assuming a solution in the form of e^(rt) and substituting it into the original equation. The resulting algebraic equation, known as the characteristic equation method, can be solved to find the roots, which determine the nature of the solution. For mechanical systems, these roots correspond to natural frequencies and damping factors, while in electrical circuits, they relate to time constants and resonant frequencies.

      Another powerful technique is the method of undetermined coefficients, which is particularly useful for equations with non-homogeneous terms. This method involves guessing the form of the particular solution based on the non-homogeneous term and then determining the coefficients to satisfy the equation. In mechanical systems, this approach can be applied to analyze forced vibrations, where external forces act on the system. For electrical circuits, it helps in studying the response to time-varying input signals.

      The variation of parameters method is a more general approach that can handle a wider range of non-homogeneous terms. This technique involves expressing the solution as a linear combination of the homogeneous solutions with variable coefficients. By solving for these coefficients, we can obtain the complete solution. This method is particularly useful in analyzing systems with complex forcing functions or when the form of the particular solution is not easily guessed.

      Laplace transforms offer a powerful alternative for solving differential equations, especially in electrical circuit analysis. By transforming the equation into the s-domain, we can convert the differential equation into an algebraic equation, solve it, and then use inverse Laplace transforms to obtain the time-domain solution. This method is particularly effective for analyzing transient responses in circuits and systems with step or impulse inputs.

      In real-world applications, these solutions provide valuable insights into system behavior. For instance, in mechanical engineering, the solution to the differential equation for a spring-mass-damper system can predict the motion of vehicle suspensions, allowing engineers to optimize ride comfort and handling. The natural frequency and damping ratio obtained from the solution help in designing structures to withstand earthquakes or vibrations from machinery.

      In electrical engineering, solutions to RLC circuit equations enable the design of filters, oscillators, and power supplies. By analyzing the frequency response derived from these solutions, engineers can create circuits that selectively amplify or attenuate specific frequency ranges, crucial for signal processing and communication systems.

      Interpreting these solutions often involves examining key parameters. In mechanical systems, the damping ratio determines whether the system is underdamped, critically damped, or overdamped, influencing the system's response to disturbances. For electrical circuits, the quality factor (Q) derived from the solution indicates the sharpness of resonance, which is vital in designing tuned circuits for radio receivers or precision measurement equipment.

      Numerical methods, such as Runge-Kutta or finite difference methods, complement analytical solutions, especially for complex systems where closed-form solutions are difficult to obtain. These computational approaches allow engineers to simulate and visualize system behavior under various conditions, aiding in design optimization and performance prediction.

      In conclusion, mastering the methods for solving second order differential equations is fundamental for understanding and designing mechanical and electrical systems. From predicting the motion of spacecraft to optimizing audio equipment, these mathematical tools provide the foundation for countless engineering applications. By combining analytical solutions with numerical simulations and real-world testing, engineers can develop innovative solutions to complex problems in fields ranging from automotive design to renewable energy systems.

      Conclusion

      Second order differential equations play a crucial role in understanding mechanical vibrations and electrical circuits. In mechanical systems, these equations model the motion of objects under the influence of forces, describing oscillations and damping effects. For electrical circuits, they represent the behavior of current and voltage in systems with inductors and capacitors. The introduction video provides a solid foundation for grasping these concepts, illustrating how these equations are derived and applied in real-world scenarios. Key points include the importance of initial conditions, natural frequency, and damping factors in determining system behavior. Understanding these applications enhances our ability to analyze and design complex mechanical and electrical systems. Students are encouraged to further explore this topic, as it forms the basis for advanced engineering concepts in fields such as control systems, signal processing, and structural dynamics. Mastering these principles opens doors to innovative solutions in various engineering disciplines.

      Applications of Second Order Differential Equations

      What are some applications of second order differential equations?

      Step 1: Introduction to Second Order Differential Equations

      Second-order differential equations are equations that involve the second derivative of a function. These equations are fundamental in various fields of science and engineering because they can describe a wide range of physical phenomena. Understanding the applications of these equations can provide insights into how they are used to model real-world systems.

      Step 2: Mechanical Vibrations

      One of the primary applications of second-order differential equations is in the study of mechanical vibrations. In engineering, these equations are used to model the behavior of mechanical systems that can oscillate, such as springs, pendulums, and other structures. The equations help in understanding how these systems respond to external forces, how they can be damped, and how they can be controlled to avoid resonant frequencies that might cause damage.

      Step 3: Electrical Circuits

      Another significant application is in electrical circuits, particularly in the analysis of RLC circuits (Resistor-Inductor-Capacitor circuits). Second-order differential equations are used to describe the voltage and current in these circuits over time. These equations help in designing circuits with desired properties, such as specific resonant frequencies or damping characteristics, which are crucial in various electronic devices and communication systems.

      Step 4: Predator-Prey Models

      Second-order differential equations are also used in ecological modeling, specifically in predator-prey models. These models describe the interactions between two species: one as a predator and the other as prey. While first-order differential equations can be used for simpler models, second-order equations allow for more complex interactions and can introduce additional variables that provide a more detailed understanding of the dynamics between species.

      Step 5: Economics and Market Equilibrium

      In economics, second-order differential equations are used to model market equilibrium and economic dynamics. These equations can describe how markets respond to changes in supply and demand over time, how economic variables such as prices and quantities adjust, and how stability can be achieved in economic systems. This application is crucial for understanding economic policies and their impacts on markets.

      Step 6: Conclusion

      In summary, second-order differential equations have a wide range of applications in various fields. From mechanical vibrations and electrical circuits to ecological models and economic systems, these equations provide powerful tools for modeling and understanding complex dynamic systems. By studying these applications, one can gain a deeper appreciation of the role of mathematics in describing and solving real-world problems.

      FAQs

      Here are some frequently asked questions about applications of second order differential equations:

      1. What are the main applications of second order differential equations?

        Second order differential equations are widely used in modeling mechanical vibrations and electrical circuits. In mechanical systems, they describe the motion of objects under the influence of forces, such as springs and dampers. In electrical circuits, they model the behavior of current and voltage in systems with inductors and capacitors.

      2. How do second order differential equations relate to mechanical vibrations?

        In mechanical vibrations, these equations describe the motion of objects in terms of displacement, velocity, and acceleration. They incorporate factors like mass, spring constant, and damping coefficient to predict how a system will oscillate or return to equilibrium when disturbed.

      3. What is the significance of damping in mechanical systems?

        Damping is crucial in controlling oscillations in mechanical systems. It determines how quickly the system returns to equilibrium after being disturbed. The damping ratio, derived from the second order differential equation, indicates whether a system is underdamped (oscillatory), critically damped (fastest return to equilibrium), or overdamped (non-oscillatory return).

      4. How are second order differential equations applied in electrical circuits?

        In electrical circuits, particularly RLC circuits, these equations describe the relationship between current, voltage, and time. They help in analyzing circuit behavior, determining resonant frequencies, and designing filters and oscillators. The equation typically involves terms related to resistance (R), inductance (L), and capacitance (C).

      5. What methods are commonly used to solve second order differential equations?

        Common methods include the characteristic equation method, method of undetermined coefficients, variation of parameters, and Laplace transforms. For complex systems, numerical methods like Runge-Kutta are often employed. The choice of method depends on the specific form of the equation and the type of system being analyzed.

      Prerequisite Topics for Applications of Second Order Differential Equations

      Understanding the applications of second order differential equations is crucial in various fields of science and engineering. However, to fully grasp this complex topic, it's essential to have a solid foundation in several prerequisite concepts. These fundamental topics provide the necessary context and tools to tackle more advanced problems in differential equations.

      One of the most important prerequisites is Newton's Second Law of motion. This fundamental principle of physics forms the basis for many applications of second order differential equations, particularly in mechanics and dynamics. By understanding how forces relate to acceleration, students can more easily interpret and solve equations describing real-world systems.

      Another crucial concept is simple harmonic motion, often studied in the context of spring systems. This topic introduces students to oscillatory behavior, which is frequently modeled using second order differential equations. Familiarity with the trigonometric functions involved in simple harmonic motion helps in visualizing and analyzing more complex oscillatory systems.

      When dealing with more advanced applications, knowledge of Laplace transforms becomes invaluable. This mathematical technique allows for the transformation of differential equations into algebraic equations, simplifying the solution process for certain types of problems. Understanding Laplace transforms can greatly enhance a student's ability to solve and analyze complex systems described by second order differential equations.

      The method of undetermined coefficients is another essential tool in the arsenal of techniques for solving second order differential equations. This method is particularly useful for finding particular solutions to non-homogeneous equations, which are common in many practical applications. Mastery of this technique allows students to approach a wide range of problems with confidence.

      By building a strong foundation in these prerequisite topics, students can more effectively navigate the complexities of second order differential equations and their applications. Each concept contributes uniquely to the overall understanding: Newton's Second Law provides physical intuition, simple harmonic motion introduces oscillatory behavior, Laplace transforms offer powerful analytical tools, and the method of undetermined coefficients expands problem-solving capabilities.

      As students progress in their studies, they'll find that these prerequisite topics are not just isolated concepts but interconnected ideas that form a comprehensive framework for understanding and applying second order differential equations. This holistic approach to learning ensures that students are well-prepared to tackle real-world problems in fields such as physics, engineering, and applied mathematics, where second order differential equations play a crucial role in modeling and analysis.

      Mechanical Vibrations:

      Hooke's Law:

      Force=-ky

      Newton's Second Law:

      Force=mass×acceleration=m×d2ydt2Force= mass \times acceleration=m \times \frac{d^2y}{dt^2}

      ⟹\Longrightarrow m×d2ydt2=−kym \times \frac{d^2y}{dt^2}=-ky

      ⟹\Longrightarrow my′′+ky=0my'' + ky =0

      Damping Force:

      Force=−cdydtForce = -c\frac{dy}{dt}

      ⟹\Longrightarrow m×d2ydt2=−cdydt−kym \times \frac{d^2y}{dt^2}=-c\frac{dy}{dt}-ky

      ⟹\Longrightarrow my′′+cy′+ky=0my''+cy'+ky=0

      Electrical Circuits:

      Using second order differential equations we are able to analyze a circuit consisting of a battery, a resistor, an inductor and a capacitor in series. Let us denote Q(t)Q(t) as the charge on the capacitor at time tt. The current is the rate of change of QQ with respect to tt. So the current of the system is equal to I=dQdtI=\frac{dQ}{dt}

      Kirchhoff's Law states that the sum of all voltage drops across a system must equal the supplied charge:

      VL+VR+VC=VbatV_L+V_R+V_C=V_{bat}

      Where VLV_L is the voltage drop across the inductor, VRV_R is the voltage drop across the resistor, VCV_C is the voltage drop across the capacitor, and VbatV_{bat} is the voltage supplied by the battery (or other electrical force).

      Faraday's Law:

      According to Faraday's Law the voltage drop across an inductor is equal to the instantaneous rate of change of current times an inductance constant, denoted by LL (measured in henry's).

      VL=L×dIdtV_L=L \times \frac{dI}{dt}

      Ohm's Law:

      From Ohm's Law the voltage drop across a resistor is equal to the resistance (measured in ohms) times the current:

      VR=IRV_R=IR

      And the voltage drop across a capacitor is proportional to the electrical charge of the capacitor times a constant of capacitance (measured in farads).

      VC=1C×Q(t)V_C=\frac{1}{C} \times Q(t)

      And let us denote the voltage from the battery as some sort of function with respect to time Vbat=E(t)V_{bat}=E(t)

      So inputting all the previously found information into Kirchhoff's Law:

      VL+VR+VC=VbatV_L+V_R+V_C=V_{bat}

      Which will become,

      LdIdt+IR+1CQ(t)=E(t)L \frac{dI}{dt}+IR+\frac{1}{C} Q(t)=E(t)

      And we know that I=dQdtI= \frac{dQ}{dt}. So the equation becomes,

      Ld2Qdt2+RdQdt+1CQ(t)=E(t)L \frac{d^2 Q}{dt^2}+R \frac{dQ}{dt}+\frac{1}{C} Q(t)=E(t)

      Which can also be written as

      LQ′′+RQ′+1CQ=E(t)LQ''+RQ'+\frac{1}{C} Q=E(t)

      Which is a second order, constant coefficient, non-homogeneous differential equation.