Editing Openai/69118e47-8450-8000-9764-f97042c547df (section)

=== Assistant: Try a power ypCx4yp  Cx^4ypCx4 since the ODE is Cauchy–Euler and r4r4r4 isn’t a root of the homogeneous part who… ===
Try a power yp=Cx4y_p = Cx^4yp=Cx4 (since the ODE is Cauchy–Euler and r=4r=4r=4 isn’t a root of the homogeneous part whose roots are 1,2,−11,2,-11,2,−1).

Compute

yp′=4Cx3,yp′′=12Cx2,yp′′′=24Cx.y_p'=4Cx^3,\quad y_p''=12Cx^2,\quad y_p'''=24Cx.yp′=4Cx3,yp′′=12Cx2,yp′′′=24Cx.
Plug into

x3y′′′+x2y′′−2xy′+2y=2x4:x^3y'''+x^2y''-2xy'+2y=2x^4:x3y′′′+x2y′′−2xy′+2y=2x4:
x3(24Cx)+x2(12Cx2)−2x(4Cx3)+2(Cx4)=(24+12−8+2)C x4=30C x4.x^3(24Cx)+x^2(12Cx^2)-2x(4Cx^3)+2(Cx^4)
= (24+12-8+2)C\,x^4=30C\,x^4.x3(24Cx)+x2(12Cx2)−2x(4Cx3)+2(Cx4)=(24+12−8+2)Cx4=30Cx4.
Set 30C x4=2x4⇒C=11530C\,x^4=2x^4\Rightarrow C=\tfrac{1}{15}30Cx4=2x4⇒C=151.

 yp(x)=x415 \boxed{\,y_p(x)=\frac{x^4}{15}\,}yp(x)=15x4