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

$y - y - 2y=4x + 5$

a particular solution is $y_p(x)=$

Explanation:

Step1: Assume the form of particular solution

Since the right - hand side is a linear function $4x + 5$, assume $y_p=Ax + B$.

Step2: Calculate the first and second derivatives

$y_p'=A$ and $y_p'' = 0$.

Step3: Substitute into the differential equation

Substitute $y_p,y_p',y_p''$ into $y''-y'-2y=4x + 5$. We get $0 - A-2(Ax + B)=4x + 5$.
Expand the left - hand side: $-A-2Ax-2B = 4x + 5$.
Group the like terms: $-2Ax-(A + 2B)=4x + 5$.

Step4: Equate the coefficients

Equate the coefficients of $x$: $-2A = 4$, so $A=-2$.
Equate the constant terms: $-A - 2B=5$. Substitute $A = - 2$ into it, we have $-(-2)-2B=5$, which simplifies to $2-2B=5$, then $-2B=3$, and $B=-\frac{3}{2}$.

Answer:

$-2x-\frac{3}{2}$