Chapter 4.5, Problem 31P

Consider the equation

$y\text{'}\text{'}-3y\text{'}-4y=2e^{-t}$ (i)

From Example 5. Recall that $y_1\left(t\right)=e^{-t}$ and $y_2\left(t\right)=e^{4t}$ are solutions of the corresponding homogeneous equation. Adapting the method of reduction of order (see the discussion preceding Problem 28 in section 4.2), seek a solution of the nonhomogeneous equation of the form $Y\left(t\right)=v\left(t\right)y_1\left(t\right)=v\left(t\right)e^{-t}$, where $v\left(t\right)$ is to be determined.

Substitute $Y\left(t\right),Y\text{'}\left(t\right)$, and $Y\text{'}\text{'}\left(t\right)$ into Eq. (i) and show that $v\left(t\right)$ must satisfy $v\text{'}\text{'}-5v\text{'}=2$.

Let $w\left(t\right)=v\text{'}\left(t\right)$ and show that $w\left(t\right)$ must satisfy $w\text{'}\text{'}-5w\text{'}=2$. Solve this equation for $w\left(t\right)$.

Integrate $w\left(t\right)$ to find $v\left(t\right)$ and then show that

$Y\left(t\right)=-\frac{2}{5}t{e}^{-t}+\frac{1}{5}{c}_1{e}^{4t}+c_2{e}^{-t}$.

The first term on the right side is the desired particular solution of the nonhomogeneous equation. Note that it is a product of $t$ and $e^{-t}$.