MAT 275 (Review)

TEST 2

Instr. S.Nikitin

Name_______________

ASU-ID#_______________

Solve the following initial value problems.

1.

$$(\frac{d}{dt})^4 y + 2 (\frac{d}{dt})^3 y + 2 (\frac{d}{dt})^2 y = 3e^t + 2te^{-t} + e^{-t}\sin(t)$$

$$y(0)=\frac{d}{dt}y(0) = (\frac{d}{dt})^2y(0)= (\frac{d}{dt})^3 y(0) = 0$$

(Use the method of undetermined coefficients).

2.

$$(\frac{d}{dt})^3 y - (\frac{d}{dt})^2 y + (\frac{d}{dt}) y - y = f(t) ,$$

where

$$f(t)=\left\{\begin{array}{ccc} 0 & \mbox{ for } & 0\le t ... ... t <2\\ 0 & \mbox{ for } & t \ge 2 \end{array} \right.$$

$$y(0)=2,\;\;\;\frac{d}{dt}y(0) = -1,\;\;\;\; (\frac{d}{dt})^2y(0)= 1$$

(Use Duhamel' integral)

3.

$$(\frac{d}{dt})^2 y + 5(\frac{d}{dt}) y + 6y = te^{-2t} + \cos(t)$$

$$y(0)=1,\;\;\;\frac{d}{dt}y(0) = 0$$

(Use the method of undetermined coefficients).

4.

$$(\frac{d}{dt})^2 y - 2(\frac{d}{dt}) y + 2y = f(t) ,$$

where

$$f(t)=\left\{\begin{array}{ccc} 0 & \mbox{ for } & 0\le t ... ...t <3 \\ 0 & \mbox{ for } & t\ge 3 \end{array} \right.$$

$$y(0)=1,\;\;\;\frac{d}{dt}y(0) = 1$$

(Use Duhamel' integral)

5. Find the general solution of the linear system

$$\frac{dx}{dt}= \left( \begin{array}{cc} 2 & 2 \\ 6 & -2 \end{array} \right) x$$