$In$ this problem you will solve the non$-$homogeneous differential equation

$y^{\text{'}\text{'}}+81y=se{c}^2(9x)$

Let $C_1$ and $C_{2}be$ arbitrary constants. The general solution $to$ the related homogeneous differential equation $y^{\text{'}\text{'}}+81y=0is$ the

function

$y_h(x)=C_1{y}_1(x)+C_2{y}_2(x)=C_1,+C_2$ help $(formulas)$

Note: The order $in$ which you enter the answers $is$ important; that $is,C_{1}f(x)+C_{2}g(x)C_{1}g(x)+C_{2}f(x).$ Therfore put $sine$

before cosine.

The particular solution $y_p(x)to$ the differential equation $y^{\text{'}\text{'}}+81y=se{c}^2(9x)isof$ the form $y_p(x)=y_1(x)u_1(x)+y_2(x)u_2(x)$

where

$u_1^{\text{'}}(x)=,$ and $u_2^{\text{'}}(x)=,$ help $(formulas)$

$It$ follows that

$u_1(x)=|$ and $u_2(x)=|$ help $(formulas)$

Thus $y_p(x)=$

help $(formulas)$

The most general solution $to$ the non$-$homogeneous differential equation $y^{\text{'}\text{'}}+81y=se{c}^2(9x)is$

$y=C_1,+C_2,,+,$ help $(formulas)$