Question

Solve the following differential equation by using the method of undetermined

coefficients:

𝑦"+4𝑦=3𝑥+𝑐𝑜𝑠(2𝑥)

Answer 1

characteristic equation:

$k^2+4=0$

$k=\pm2i$

$y_h=c_1cos2x+c_2sin2x$

particular solution:

$y_{p1}=Ax+B$

$A=3/4,B=0$

$y_{p2}=Axcos2x+Bxsin2x$

$y'_{p2}=Acos2x+Bsin2x+x(-2Asin2x+2Bcos2x)$

$y''_{p2}=-2Asin2x+2Bcos2x-2Asin2x+2Bcos2x+x(-4Acos2x-4Bsin2x)$

$-2Asin2x+2Bcos2x-2Asin2x+2Bcos2x+x(-4Acos2x-4Bsin2x)+$

$+4(Axcos2x+Bxsin2x)=cos2x$

$-4A=0\implies A=0$

$4B=1\implies B=1/4$

$y=c_1cos2x+c_2sin2x+3x/4+xsin2x/4$