Question

find the particular integral of (D^2-3D+2)y=e^x​

Answer 1

Answer:

You are almost there, but note that W(x)=e−3x

PI=−e−2x∫e2xeexdx+ex∫exeexdx

Let ex=t⟹dx=dtt, then

PI=−e−2x∫tetdt+e−x∫etdt

PI=−e−2xet(t−1)+e−xet=e−2x+ex.

It's Euler-Cauchy's equation hidden. To simplify substitute u=ex then equation becomes:

(u2D2+4uD+2)y=eu

Try y=xm

m2+3m+2−0⟹m=−1,−2

⟹Sy={1u,1u2}

(D2+4uD+2u2)y=euu2

The Wronskian is |W|=−1u4 And the particular solution is:

yp=euu2

Substitute back u=ex

Step-by-step explanation: