Order and solutions to differential equations

  1. Differential Equations Overview
  2. Notation and Order of a Differential Equation
  3. Solution to a Differential Equation
  4. Particular Solutions
  1. Finding the Order of a Differential Equation
    What is the order for the following differential equations?
    1. y=t2+7y''' = {t^2} + 7
    2. y(t)+6y(t)+8y(t)=ln(t)y''( t ) + 6y'( t ) + 8y( t ) = ln( t )
    3. 5dydt=csc(4t)+d2ydt25\frac{{dy}}{{dt}} = \csc \left( {4t} \right) + \frac{{{d^2}y}}{{d{t^2}}}
    4. x3d3ydx3+x2d2ydx2+xdydx=1{x^3}\frac{{{d^3}y}}{{d{x^3}}} + {x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} = 1
  2. Verifying Solutions
    Show that the following functions is a solution to the differential equation:
    1. y=cos(5t)y = \cos( 5t ) is a solution to y+25y=y'' + 25y = 0
    2. y=Ccos(5t)y = C\cos \left( {5t} \right) is a solution to y+25y=y'' + 25y = 0 where CC is a constant
    3. y=Cexy = C{e^x} is a solution to 3y+y+y3y=2Cex3y + y' + y'' - 3y''' = 2C{e^x} where CC is a constant
    4. y=x44+x2y = \frac{{{x^4}}}{4} + {x^2} is a solution to y=6xy''' = 6x
  3. Finding a Particular Solution
    You are given the general solution as well as the initial condition. Find the particular solution which suits the following initial conditions:
    1. 6x3+9y2=C6{x^3} + 9{y^2} = C where y(0)=3y\left( 0 \right) = 3
    2. Ce3t=y2C{e^{3t}} = {y^2} where y(1)=1y\left( 1 \right) = 1
  4. Integrating to Find the General Solution
    Find the general solution of the differential equation  dydx=xex2\;\frac{{dy}}{{dx}} = x{e^{{x^2}}}.
    Topic Notes
    In this lesson, we will look at the notation and highest order of differential equations. To find the highest order, all we look for is the function with the most derivatives. After, we will verify if the given solutions is an actual solution to the differential equations. We do this by simply using the solution to check if the left hand side of the equation is equal to the right hand side. Lastly, we will look at an advanced question which involves finding the solution of the differential equation.

    Differential Equations

    1. Introduction to Differential equations

    a. Order and solutions to differential equations
    During this lesson you will learn what differential equations are along with how to classify them depending on the characteristics they display. Therefore, you will take a look into the notation and order of a differential equation.
    This lesson is an introduction to concepts we will be using throughout the whole Differential Equations course.

    2. First Order Differential Equations

    a. Slope fields
    Slope fields (also usually called vector fields) provide us with visual representations of differential equations which aid us in finding the solutions for such equations. During this lesson, you will look into examples of first order differential equations where the use of a graphic representation is needed since an explicit solution is either complicated or not possible to obtain.

    b. Separable Equations
    Separable equations are differential equations which can be solved through the method of separation of variables. Throughout this lesson you will learn to identify and work through the algebraic operations that will allow you to isolate same-variable terms on each side of the equal sign of separable differential equations.

    c. Exact differential equations
    An exact differential equation has a solution equal to a constant value. You will see throughout this lesson that you can identify exact equations by looking at a mixed partial derivatives condition, where the 2nd partial derivatives of the solution (once for each of the variables, continuously differentiated in different order) are equal to each other, creating a precedent which helps solve first order differential equations in a relatively simple manner.

    d. Integrating factor technique
    The integrating factor is a technique we use to produce a simpler-to-solve differential equation on a complicated problem. By multiplying the given equation by a function of the independent variable in use, we can produce a new expression which can be solved effectively using easier methods such as exact equations. Although finding integrating factors can make our calculations robust, it is still an important tool to learn for solving differential equations.

    e. Bernoulli equations
    A Bernoulli differential equation is a nonlinear first order differential equation which has known exact solutions and therefore, it can be reduced to a linear first order differential expression to be solved through methods we already know, such as the integrating factor method with exact equations. During this lesson you will learn how to identify Bernoulli equations and steps on how to solve them.

    f. Interval of validity
    The interval of validity is a tool that allows us to infer if the solutions we have obtained for first order differential equations are viable. The practicality of this tool comes from the fact that we can make our well funded inferences sometimes just by looking into the initial condition of the problem we are solving. Thus, on this lesson we will explain what is an interval of validity, the uniqueness theorem and methods on how to calculate them while solving differential equations.

    g. Modeling with differential equations
    In this section, we will try to apply differential equations to real life situations. For each question we will look how to set up the differential equation. Afterwards, we will find the general solution and use the initial condition to find the particular solution. Depending on the question, we will even look at behaviours of the differential equation and see if it is applicable to real life situations. For example, one can notice that integrating the area of a sphere actually gives the volume of a sphere!

    h. Equilibrium solutions
    An equilibrium solution for a given function represents the equilibrium point in the graphic representation of the function, which can be seen as the region where the function line runs horizontally thus having a value of zero for the slope of its tangent line. Equilibrium solutions are important when solving autonomous differential equations.

    i. Eulers method
    The Eulers method is an approximation method for solving differential equations when a perfect solution for the given equations cannot be found. As you will be able to see throughout our lesson, Eulers linear approximation formula allows us to get as close as possible to the solution of an unconventional problem.

    3. Second Order Differential Equations

    a. Homogeneous linear second order differential equations
    So far we have been introduced to first order linear differential equations and the methods to find their solutions. Throughout this lesson we will follow the properties we learnt at the beginning of the chapter about notation and order of differential equations, to identify and work through homogeneous linear second order differential equations.

    b. Characteristic equation with real distinct roots
    On this lesson we will talk about how we can convert a second order linear differential equation into a second degree algebraic equation through a characteristic equation and solve for its roots in order to use that information to solve the differential equation.
    A similar process will be used for the next two lessons, but the special case for today is that we will find real distinct roots.

    c. Characteristic equation with complex roots
    Using similar methodology than the one described in the past lesson, we will solve second order differential equations using a characteristic equation to transcribe them into an algebraic quadratic equation from which we can obtain its roots. For this lesson, the roots will be composed of complex numbers which are part real and part imaginary numbers, and so, the result are complex roots.

    d. Characteristic equation with repeated roots
    The use of a characteristic equation to solve second order linear differential equations by converting them into second degree algebraic expressions has been explained for the past two lessons. On this lesson, we will see the special case in which the roots happen to be repeated roots.

    e. Reduction of order
    The method of reduction of order reduces (simplifies) a second order differential equation given into first order differential equations which have to be solved one after the other. There is not a unique reduction of order formula that works for every scenario, so the reduction of order method is actually a combination of methods we have seen in past lessons to solve differential equations. Thus, through this lesson you will gain practice at identifying the right tools to use while reducing and finding the solutions to differential equations using this method.

    f. Wronskian
    The wronskian is a tool used to prove if the solutions found for a differential equation are a set of fundamental solutions. And so, this technique allows you to prove that the computed solutions for a differential equation satisfy the initial conditions of the problem and, when combined, can give the most efficient final general solution to the problem. During this lesson you will learn how to compute the wronskian matrix and the information you can obtain from its determinant.

    g. Method of undetermined coefficients
    The method of undetermined coefficients is a technique used to find the solution to nonhomogeneous second order differential equations. As you will see, the name of this technique comes from the need to find the value of the undetermined coefficients from the complementary solution to the differential equation.

    h. Applications of second order differential equations
    This section will concentrate on showing the applications of second order differential equations, and the methods we use to find their solutions, through different areas of study in engineering and physics. You will be using methods seen in the past lessons regarding this chapter, thus, make sure to go back to the lessons and review them whenever you think is needed.

    4. Laplace Transforms

    a. Introduction to the Laplace transform
    The Laplace transform is an operator which allows us to simplify heavy calculus-based computations into a more algebra-based process while solving differential equations. Since this lesson is an introduction to the Laplace transformation only, you will learn how to calculate the transform of a few widely used functions in order to produce a Laplace transform table that will helps us in later lessons while solving problems.

    b. Calculating Laplace transforms
    During this lesson we will learn the method of comparison with the table of Laplace transforms in order to solve Laplace transformations. You have had an introduction to the Laplace transform in the past lesson, and so this time you will learn the simplest and most efficient method to calculate any Laplace transforms using those past results.

    c. Inverse Laplace transforms
    The inverse Laplace transform is the inverse operation from the Laplace transform. On this lesson you will use the table of Laplace transforms from the past two lessons in order to do some reverse engineering on the functions given to solve. In other words, you will be given Laplace transforms (functions in terms of s) and you will compare these functions with the ones found in the table and compute the functions in terms of t where they came from.

    d. Solving differential equations with the Laplace transform
    Using the table of Laplace transforms and the methodology from the past three lessons, we use our gathered knowledge to now solve differential equations using the Laplace transformation. You will notice how the use of Laplace transforms to solve the differential equations change the computations from calculus to algebraic techniques such as partial fractions.

    e. Step functions
    Unit step function or Heaviside function is a functions that has a constant value of zero up to a certain point on t (the horizontal axis) and then it jumps to a value of 1. This lesson will prepare you as an introduction to step functions, so you can use them combined with Laplace transforms to solve differential equations problems in the next lesson.

    f. Solving differential equations with step functions
    During this lesson we will use the knowledge acquired throughout our past lesson about the unit step function or Heaviside function, and the computation of the Laplace transform of such step functions using a set of well known formulas in order to facilitate the solution of differential equations. The lesson contains a re-introduction to step functions and one example only, but the methodology used during its calculation is extensively explained, including a list of steps, and other references of study provided.

    g. Dirac Delta function
    The Dirac Delta function provides a graphic representation of an instantaneous impulse from an information-filled signal. Also called the unit impulse function, the Dirac Delta function helps us mathematically understand the behaviour of instantaneous bits of information in otherwise empty or isolated (no other function is found) reference frames.

    h. Convolution integral
    The convolution integral method provides us with a simpler approach to solve over complicated inverse Laplace transforms. The properties of the convolution integral allow us to combine two mathematical functions into one which can be used later to solve different kinds of problems, many of them widely found in our internet filled days, making convolution one of the most important techniques used in signal processing for our technological world.
    We say that:
    y(x)=dydxy' (x) = \frac{dy}{dx} or y(t)=dydty'(t) = \frac{dy}{dt}

    1. y(x)y'(x) is the first derivative of the function y in terms of xx.
    2. y(t)y'(t) is the first derivative of the function y in terms of tt.