MAT 274 (Review)

TEST 3

Instr. S.Nikitin

Name_______________

ASU-ID#_______________

I. Solve the following initial value problems.

1.

$$\frac{d}{dt}y = \left( \begin{array}{cc} -8 & 18 \\ -3... ...\left( \begin{array}{c} e^{-t}\\ 1 \end{array} \right)$$

$$y(0)=\left( \begin{array}{c} 4\\ 1 \end{array} \right)$$

2.

$$\frac{d}{dt}y = \left( \begin{array}{cc} -2 & 1 \\ 1 &... ...eft( \begin{array}{c} e^t\\ e^{-t} \end{array} \right)$$

$$y(0)=\left( \begin{array}{c} 4\\ 1 \end{array} \right)$$

II. Find the general solution, sketch the phase portrait.

a.

$\frac{d}{dt}y = \left( \begin{array}{cc} -3 & 1 \\ -5 & 1 \end{array} \right)$

b.

$\frac{d}{dt}y = \left( \begin{array}{cc} -1 & 1 \\ -5 & 1 \end{array} \right)$

c.

$\frac{d}{dt}y = \left( \begin{array}{cc} -2 & \frac13 \\ -3 & -2 \end{array} \right)$

III. Find the solution in the integral form (you do not need to calculate the integrals).

$$\frac{d}{dt}y = \left( \begin{array}{cc} 0 & 1 \\ -25 ... ...in{array}{c} \sqrt{sin^2(t)} \\ t \end{array} \right)$$

$$y(0)=\left( \begin{array}{c} 1\\ 1 \end{array} \right)$$

IV. Find the solution with the help of Laplace transforms.

$$\ddot y(t) + 3 \dot y + 2 y = \sin(t)$$

$$y(0)=-1,\;\;\dot y(0)=1$$