Chapter 1.1, Problem 16E

(a)

To determine

The system of first order equations equivalent to the second order differential equation $\frac{d^{2}x}{d{t}^2}-\frac{dx}{dt}-6x=0$.

(b)

To determine

The system of first order equations equivalent to the second order differential equation $4\frac{d^{2}x}{d{t}^2}+4\frac{dx}{dt}+37x=0$.

(c)

To determine

The system of first order equations equivalent to the second order differential equation $L\frac{d^{2}x}{d{t}^2}+g\sin x=0$ where $L$, $g$ are positive constants and $x=x\left(t\right)$.

(d)

To determine

The system of first order equations equivalent to the second order differential equation $\frac{d^{2}x}{d{t}^2}-\mu \left(1-x^2\right)\frac{dx}{dt}+x=0$ and $\mu >0$.

(e)

To determine

The system of first order equations equivalent to the second order differential equation $t\frac{d^{2}x}{d{t}^2}-\left(b-t\right)\frac{dx}{dt}-ax=0$, where $a$ and $b$ are constant.