Exercise 4

Change each of the following second-order initial value problems into a system of equations.

1. $$\begin{aligned} x'' + 6x' - 2x & = 0 \\ x(0) & = 1 \\ x'(0) & = 1 \end{aligned}$$

2. $$\begin{aligned} y'' + y' - 2y & = 0 \\ y(0) & = 2 \\ y'(0) & = 0 \end{aligned}$$

3. $$\begin{aligned} \frac{d^2x}{dt^2} + \frac{dx}{dt}' + 4x & = \sin t \\ x(0) & = 4 \\ x'(0) & = -3 \end{aligned}$$

4. $$\begin{aligned} x'' + tx' - 2 & = 0 \\ x(0) & = 0 \\ x'(0) & = 1 \end{aligned}$$