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# Applications of second order differential equations

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### Applications of second order differential equations

#### Lessons

__Mechanical Vibrations:__

__Hooke’s Law:__

*Force=-ky*

__Newton’s Second Law:__

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

$\Longrightarrow$ $m \times \frac{d^2y}{dt^2}=-ky$ $\Longrightarrow$ $my'' + ky =0$__Damping Force:__

__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 $Q(t)$ as the charge on the capacitor at time $t$. The current is the rate of change of $Q$ with respect to $t$. So the current of the system is equal to $I=\frac{dQ}{dt}$

__Kirchhoff’s Law__ states that the sum of all voltage drops across a system must equal the supplied charge:

Where $V_L$ is the voltage drop across the inductor, $V_R$ is the voltage drop across the resistor, $V_C$ is the voltage drop across the capacitor, and $V_{bat}$ 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 $L$ (measured in henry’s).

__Ohm’s Law:__

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

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

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

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

Which will become,

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

Which can also be written as

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

- 1.a)What are some applications of second order differential equations?b)Mechanical Vibrations and Dampening Forcesc)Damping Forces on Mechanical Vibrationsd)Electrical Circuits
- 2.

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 $t$?**Mechanical Vibrations** - 3.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.a)What is the position of the mass at time $t$ if the hydraulic shock has a damping constant of $c=50$, with an initial positions of 0.75 meters, and an initial velocity of 0 $m/s$?b)What is the position of the mass at time $t$ if the hydraulic shock has a damping constant of $c=40$, with an initial position of 0.5 meters, and an initial velocity of 5 $m/s$?c)What is the position of the mass at time $t$ if the hydraulic shock has a damping constant of $c=20$, with an initial position of 0.3 meters, and an initial velocity of -0.3 $m/s$?
- 4.
**Electrical Circuits**Find the charge at time $t$ for an electrical circuit with a resistor that has a resistance of $R=14 \Omega$ , an inductor with $L=2H$, a capacitor with $C=0.05 F$, and a battery with charge $E(t)=8$$\sin(2t)$. The initial charge is $V=\frac{22}{29}$ coulombs, and the initial current is $I=\frac{6}{29}$ amps.