Math 555: Differential Equations I


Final Exam

Review Guide


This page contains questions similar in spirit to those that will appear on the Final Exam. The Final Exam will cover material from the entire course. Students should also study the past Good Problems.

The Final Exam is scheduled for Thursday, 25 July 2019 and Friday, 26 July 2019 at 9:50 am in our usual classroom. There are no make-ups allowed on the final exam, and no extra time will be given. Please make sure that you are on time.

The final exam will be formatted as follows:

Each day's portion of the exam will consist of 5 quesions: one from each of the five groups of review questions listed below. The questions will be changed slightly, but the spirit will remain the same. Each question will be worth 20 points.

Students will be allowed to use two index cards (60 square inches of total surface area) of their own hand-written notes each day.

Extra Credit:
Students who wish to do so may submit solutions to this review guide for extra credit. The solutions must be written neatly on the student's own paper and submitted

by 10:50 am on Wednesday, July 24.



Group I

1. Solve the equation \( (2y+1)dx + (x - \tfrac{y}{x})dy = 0 \).

2. Find the solution to the initial value problem and determine the domain of definition of the solution.

\[ \begin{cases} y' = 2xy^2,\\ y(0) = -1. \end{cases} \]
3. Consider the initial value problem.

\[ \begin{cases} \ln(t)y' - \sin(t)y = t^2 + e^{2t}, \\ y(1) = -\pi. \end{cases} \]
Apply the Fundamental Existence and Uniqueness Theorem to show that a solution exists. Can you determine the domain of the solution without solving the IVP explicitly? If so, what is it? If not, why?

4. Consider the initial value problem

\[ \begin{cases} \dfrac{dy}{dt} = \dfrac{t + ty^2}{e^t}, \\ y(0) = 1. \end{cases} \]
Apply the Fundamental Existence and Uniqueness Theorem to show that a solution exists. Then solve the IVP using your favorite method. What is the domain of definition of the solution function?


Group II

5. Solve the initial value problem

\[ \begin{cases} y'' + 4y' + 3y = 0, \\ y(0) = 2, y'(0) = -1. \end{cases} \]
6. Consider the equation \( xy'' + (3 -2x)y' - 4y = 0\), \(x > 0\). If \(y_1(x) = x^{-2}\) is a solution of the equation, use the method of reduction of order to find the general solution.

7. Solve the initial value problem. \[ \begin{cases} y'' - 2y' + 5y = 8\sin x - 4\cos x, \\ y(0) = 3,\ \ \ y'(0) = 9. \end{cases} \]
8. Suppose \(g = g(t)\) is an integrable function. Write down an integral formula for the solution to the non-homogeneous differential equation

\[ y''' + 3y'' + y' - 5y = g(t) \]

Group III

9. Consider the equation \(y'' - 4y' + 4y = x^2e^x + 4xe^x\sin x + 6\cos x + 3\). Determine a suitable form of a particular solution by the method of undetermined coefficients. Do not evaluate the coefficients.

10. It is known that \(x^2\) and \(x^2\ln x\) are solutions of the homogeneous equation associated to the nonhomogeneous equation \[ x^2y'' -3xy' + 4y = x^2\ln x. \] Use the method of variation of parameters to find a particular solution to the nonhomogeneous equation.

11. Find the general solution to the non-homogeneous Euler equation using your favorite method. \[ x^2y'' - 4xy' + 6y = 2x^3\ln(x), \ \ \ x > 0. \]

12. Apply Abel's Theorem to find an expression for the Wronksian of the fundamental solution set without solving the differential equation.

\[ t^2y'' - 5ty' + 6t^2y = 0. \]


Group IV

13. Given the equation \(x^2y'' + 8xy' + 12(1+x)y = 0\). Then \(x = 0\) is a regular singular point. Show a correct form of the series solutions to the equation.

14. Use the power series method to find a fundamental set for the equation \(y'' - 3xy' + y = 0\). Determine the first three terms in each of the two solutions that form the fundamental set.

15. You wish to find a series solution to the initial value problem,

\[ \begin{cases} (1 + x^2)y'' - 3xy' + \dfrac{17}{x}y = 0, \\ y(1) = 2. \end{cases} \]
Without solving the problem, determine a lower bound on the radius of convergence of the series solution.

16. Determine the radius and interval of convergence of the power series,

\[ \varphi(x) = \sum_{n=0}^\infty \dfrac{3n}{n^2 - 3} (x - 3)^{2n}. \]


Group V

17. Find the inverse Laplace transform for \(F(s) = \dfrac{e^{-2s}(s-1)} {s^2+2s+5}\).

18. Use the Laplace transform to solve the problem. \[\begin{cases} y'' + 2y' + 5y = 0 \\ y(0) = 2,\ \ \ y'(0) = -1 \end{cases}\]

19. Use the Laplace transform to solve the problem. \[\begin{cases} y'' + 4y = u_\pi (t) \\ y(0) = y'(0) = 0. \end{cases}\]
20. Use the definition of Laplace transform to compute \(\mathcal{L}\left\{u_c(t)\right\}\). You must use the definition of Laplace transform to receive credit. Be sure to treat the improper integral properly.



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