Answer to Question #295146 in Differential Equations for Bala

Question #295146

Find a power series solution pf xy'=y

1
Expert's answer
2022-02-08T17:56:01-0500

CORRECTED SOLUTION.

Let y=n=0+cnxny=\sum\limits_{n=0}^{+\infty}c_nx^n. Then xy=xn=0+ncnxn1=n=0+ncnxnxy'=x\sum\limits_{n=0}^{+\infty}nc_nx^{n-1}=\sum\limits_{n=0}^{+\infty}nc_nx^n.

Since xy=yxy'=y, it must be ncn=cnnc_n=c_n for all n=0,1,2,n=0,1,2,\dots

This means cn=0c_n=0 for all n1n\ne 1 and c1c_1 can be arbitrary.


Answer. y=c1xy=c_1x.


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