Question

find a particular solution $y_p$ of the following equation using the method of undetermined coefficients. primes denote the derivatives with respect to $x$.
$y + 13y = 5e^{9x}$
a particular solution is $y_p(x)=$

Explanation:

Step1: Assume the form of particular solution

Since the right - hand side is $5e^{9x}$, assume $y_p = Ae^{9x}$, where $A$ is a constant to be determined.

Step2: Calculate the first and second derivatives

The first derivative $y_p'=9Ae^{9x}$, and the second derivative $y_p'' = 81Ae^{9x}$.

Step3: Substitute into the given differential equation

Substitute $y_p$, $y_p'$ and $y_p''$ into $y'' + 13y=5e^{9x}$. We get $81Ae^{9x}+13(Ae^{9x}) = 5e^{9x}$.
Combining like terms, $(81A + 13A)e^{9x}=5e^{9x}$, so $94Ae^{9x}=5e^{9x}$.

Step4: Solve for $A$

Since $e^{9x}
eq0$ for all $x$, we can divide both sides by $e^{9x}$ to get $94A = 5$. Then $A=\frac{5}{94}$.

Answer:

$y_p(x)=\frac{5}{94}e^{9x}$