Answer to Question #283565 in Differential Equations for Palli

Question #283565

Solve (𝐷


2 − 3𝐷 + 2)𝑦 = 𝑥


2 + sin 𝑥

1
Expert's answer
2021-12-30T05:41:45-0500

characteristic equation:

k23k+2=0k^2-3k+2=0

k=3±982k=\frac{3\pm \sqrt{9-8}}{2}

k1=1,k2=2k_1=1,k_2=2

complementary solution:

yc=c1ex+c2e2xy_c=c_1e^x+c_2e^{2x}


for particular solution:

yp1=Ax2+Bx+Cy_{p1}=Ax^2+Bx+C

2A3(2Ax+B)+2(Ax2+Bx+C)=x22A-3(2Ax+B)+2(Ax^2+Bx+C)=x^2

A=1/2A=1/2


2A3B+2C=02A-3B+2C=0

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

C=(3B2A)/2=(9/21)/2=7/4C=(3B-2A)/2=(9/2-1)/2=7/4


yp2=Acosx+Bsinxy_{p2}=Acosx+Bsinx

AcosxBsinx3(Asinx+Bcosx)+2(Acosx+Bsinx)=sinx-Acosx-Bsinx-3(-Asinx+Bcosx)+2(Acosx+Bsinx)=sinx

A4B=0A-4B=0

3A+B=13A+B=1

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


y=yc+yp1+yp2=c1ex+c2e2x+x2/2+3x/2+7/4+4cosx/13+sinx/13y=y_c+y_{p1}+y_{p2}=c_1e^x+c_2e^{2x}+x^2/2+3x/2+7/4+4cosx/13+sinx/13


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment