Third order Cauchy-Euler differential equation Solve: x^{3}y'''-x^{2}y''+2xy'-2y=x^{3}

Zerrilloh6

Zerrilloh6

Answered question

2022-01-21

Third order Cauchy-Euler differential equation
Solve: x3yx2y+2xy2y=x3

Answer & Explanation

zurilomk4

zurilomk4

Beginner2022-01-21Added 35 answers

I just wanted to try using Laplace:
x3yx2y+2xy2y=x3
yy+2y2y=e3t
See above link for substitution details: x=et
Hence forth, y
accimaroyalde

accimaroyalde

Beginner2022-01-22Added 29 answers

x3yx2y+2xy2y=x3
Consider the substitution due to Euler:
ez=x
Youll
RizerMix

RizerMix

Expert2022-01-27Added 656 answers

I can't help but feel like the answers given in this thread are way too complicated and although they work, the easier solution is just to look at the repeated root and realize that two of the linearly independent solutions are y1=c1x and y2=c2xln(x). Why? It's easy to prove with reduction of order for a 2nd order linear homogeneous cauchy euler equation. I know your question is 4 years old, so I won't bother typing up a proof for nothing, but if anyone else stumbles upon this thread, you can message me and I'll explain in more detail.

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