Question

Solve (𝐷

2 − 3𝐷 + 2)𝑦 = 𝑥

2 + sin 𝑥

Answer 1

characteristic equation:

$k^2-3k+2=0$

$k=\frac{3\pm \sqrt{9-8}}{2}$

$k_1=1,k_2=2$

complementary solution:

$y_c=c_1e^x+c_2e^{2x}$

for particular solution:

$y_{p1}=Ax^2+Bx+C$

$2A-3(2Ax+B)+2(Ax^2+Bx+C)=x^2$

$A=1/2$

$2A-3B+2C=0$

$-6A+2B=0\implies B=3/2$

$C=(3B-2A)/2=(9/2-1)/2=7/4$

$y_{p2}=Acosx+Bsinx$

$-Acosx-Bsinx-3(-Asinx+Bcosx)+2(Acosx+Bsinx)=sinx$

$A-4B=0$

$3A+B=1$

$B=1/13,A=4/13$

$y=y_c+y_{p1}+y_{p2}=c_1e^x+c_2e^{2x}+x^2/2+3x/2+7/4+4cosx/13+sinx/13$