Real Eigenvalues — In this section we will solve systems of two linear differential equations in which the eigenvalues are distinct real numbers. We will also show how to sketch phase portraits associated with real distinct eigenvalues saddle points and nodes. Complex Eigenvalues — In this section we will solve systems of two linear differential equations in which the eigenvalues are complex numbers. This will include illustrating how to get a solution that does not involve complex numbers that we usually are after in these cases.
We will also show how to sketch phase portraits associated with complex eigenvalues centers and spirals. Repeated Eigenvalues — In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated double in this case numbers. This will include deriving a second linearly independent solution that we will need to form the general solution to the system.
We will also show how to sketch phase portraits associated with real repeated eigenvalues improper nodes. Nonhomogeneous Systems — In this section we will work quick examples illustrating the use of undetermined coefficients and variation of parameters to solve nonhomogeneous systems of differential equations. Laplace Transforms — In this section we will work a quick example illustrating how Laplace transforms can be used to solve a system of two linear differential equations.
In particular we will look at mixing problems in which we have two interconnected tanks of water, a predator-prey problem in which populations of both are taken into account and a mechanical vibration problem with two masses, connected with a spring and each connected to a wall with a spring.
Power Series — In this section we give a brief review of some of the basics of power series. Taylor Series — In this section we give a quick reminder on how to construct the Taylor series for a function. Series Solutions — In this section we define ordinary and singular points for a differential equation. We also show who to construct a series solution for a differential equation about an ordinary point. The method illustrated in this section is useful in solving, or at least getting an approximation of the solution, differential equations with coefficients that are not constant.
Note that while this does not involve a series solution it is included in the series solution chapter because it illustrates how to get a solution to at least one type of differential equation at a singular point.
Linear Homogeneous Differential Equations — In this section we will extend the ideas behind solving 2 nd order, linear, homogeneous differential equations to higher order. We will also need to discuss how to deal with repeated complex roots, which are now a possibility.
In addition, we will see that the main difficulty in the higher order cases is simply finding all the roots of the characteristic polynomial. Undetermined Coefficients — In this section we work a quick example to illustrate that using undetermined coefficients on higher order differential equations is no different that when we used it on 2 nd order differential equations with only one small natural extension.
Variation of Parameters — In this section we will give a detailed discussion of the process for using variation of parameters for higher order differential equations. We will also develop a formula that can be used in these cases. We will also see that the work involved in using variation of parameters on higher order differential equations can be quite involved on occasion. Laplace Transforms — In this section we will work a quick example using Laplace transforms to solve a differential equation on a 3 rd order differential equation just to say that we looked at one with order higher than 2 nd.
As we will see they are mostly just natural extensions of what we already know who to do. Series Solutions — In this section we are going to work a quick example illustrating that the process of finding series solutions for higher order differential equations is pretty much the same as that used on 2 nd order differential equations.
We will also work a few examples illustrating some of the interesting differences in using boundary values instead of initial conditions in solving differential equations. Eigenvalues and Eigenfunctions — In this section we will define eigenvalues and eigenfunctions for boundary value problems.
We will work quite a few examples illustrating how to find eigenvalues and eigenfunctions. In one example the best we will be able to do is estimate the eigenvalues as that is something that will happen on a fairly regular basis with these kinds of problems. Periodic Functions and Orthogonal Functions — In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions.
The results of these examples will be very useful for the rest of this chapter and most of the next chapter. We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function.
We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function. Fourier Series — In this section we define the Fourier Series, i. We will also work several examples finding the Fourier Series for a function. Convergence of Fourier Series — In this section we will define piecewise smooth functions and the periodic extension of a function.
In addition, we will give a variety of facts about just what a Fourier series will converge to and when we can expect the derivative or integral of a Fourier series to converge to the derivative or integral of the function it represents. The Heat Equation — In this section we will do a partial derivation of the heat equation that can be solved to give the temperature in a one dimensional bar of length L.
In addition, we give several possible boundary conditions that can be used in this situation. We also define the Laplacian in this section and give a version of the heat equation for two or three dimensional situations. The Wave Equation — In this section we do a partial derivation of the wave equation which can be used to find the one dimensional displacement of a vibrating string.
In addition, we also give the two and three dimensional version of the wave equation. Terminology — In this section we take a quick look at some of the terminology we will be using in the rest of this chapter. Solve the differential equation for your bank balance given as follows. The following differential equation represents the change in population as a function of time. What does the 0. Which of the following might be a differential equation for how the velocity of a kite depends on its position and the time?
What is the equation for L as a function of x? Which of the following statements is TRUE? Page 3 Question 11 How fast is the surface area of the balloon increasing when its radius is 16cm? Which of the following is NOT a differential equation? Page 4 Question 16 Imagine you have a sheet cake with a width of 12 inches and a length of 12 inches. One ounce is about 1 cubic inch of icing. You have a tank that is shaped like a prism that is on its side.
The triangular base of the tank is a right triangle of height 10 m and width 10 m. See the image below. If the tank is being filled at a rate of 30 cubic meters per minute, how quickly is the height changing? Which of the following might describe how the population P changes as a function of time t? Page 5 Question 21 The number of widgets in your stock inventory N changes over time.
You sell 5 times your stock inventory of widgets an hour and generate widgets at a rate given by the equation g t. Which equation below describes this relationship? Previous Page Next Page. Create an account today. Browse Browse by subject. Email us if you want to cancel for any reason. Start your FREE trial. Differential Equations 5th Edition Edit editions.
Consider the differential equation as shown below: Consider the initial value problem, Recollect that the solution of the initial value problem with is …… 1 Here , , , and. Start with To find , let in 1 , then. Continue the calculations, Generate the sequence of partial sums of power series of Therefore, the exponential series is. View a full sample. C Henry Edwards Authors: Need an extra hand? Browse hundreds of Math tutors. How is Chegg Study better than a printed Differential Equations 5th Edition student solution manual from the bookstore?
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See All Differential Equations Homework Students normally study the topic of differential equations after several semesters of calculus, at which time it becomes a natural extension of the former. Simply defined, differential equations are equations that contain one or more differentials of functions.
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