Differential equations -- Spring 2017 -- final exam

MATH 335 -- Spring 2017 -- final exam

Name:_________________________ $\qquad$ 8 problems, 10 points each

• Write the problems in the given order, each problem on a separate page.
• Show enough work to justify your answers.

$\bullet$ 1. Solve by separating variables: $$y'=xy.$$

$\bullet$ 2. Solve by the method of integrating factor: $$y'=y/x.$$

$\bullet$ 3. Carry out $n=4$ steps of Euler's method with $h=.5$ for the following initial value problem: $$y'=y-x,\ y(0)=2.$$

$\bullet$ 4. Provide the definition of the uniqueness property and sketch an example of a solution set without it.

$\bullet$ 5. A sketch of the solution set of an ODE $y'=f(t,y)$ is shown below. What can you say about the sign of $f$?

$\bullet$ 6. Find the eigenvalues and the eigenvectors of the following matrix: $$F=\left[\begin{array}{c}2&1\\1&2\end{array}\right].$$

$\bullet$ 7. Sketch the trajectories of a system of linear ODEs $X'=FX$ if the matrix $F$ has these pairs of eigenvalues and eigenvectors: $$\lambda_1=2,\ V_1=\left[\begin{array}{c}1\\0\end{array}\right] \text{ and } \lambda_2=-1,\ V_2=\left[\begin{array}{c}1\\1\end{array}\right].$$

$\bullet$ 8. Write the general solution of a system of linear ODEs $X'=FX$ if the matrix $F$ has these pairs of eigenvalues and eigenvectors: $$\lambda_1=i,\ V_1=\left[\begin{array}{c}1\\i\end{array}\right] \text{ and } \lambda_2=-i,\ V_2=\left[\begin{array}{c}1\\-i\end{array}\right],$$ and provide one non-trivial real solution of the system.