Chapter 2.1, Problem 38P

(a)

To determine

To prove: The function $y=A{e}^{-{\displaystyle \int p\left(t\right)}dt}$ is a solution of the differential equation $y^{\prime }+p\left(t\right)y=g\left(t\right)\text{ if }g\left(t\right)=0$ where $A$ is a constant.

(b)

To determine

To prove: The solution $y=A\left(t\right)e^{-{\displaystyle \int p\left(t\right)}dt}$ of the differential equation $y^{\prime }+p\left(t\right)y=g\left(t\right)$ where $g\left(t\right)$ not zero everywhere must satisfy the condition $A^{\prime }\left(t\right)=g\left(t\right)e^{{\displaystyle \int p\left(t\right)}dt}$.

(c)

To determine

To show: The solution obtained in the above manner satisfies the equation 33 by computing the function $A\left(t\right)$.