solve the CauchyEuler differential equation by variation of

solve the Cauchy-Euler differential equation by variation of parameters. x^2+y\'\'+xy\'-y=lnx

Solution

Since, the RHS has Inx we have to take y=x-AIn(X), where A is the constant we need to find.

Substituting this in the above differential eqn. we have y\' = 1-A/x and y\'\' = A/x^2.

x^2*(A/x^2) + (1-A/x)x -( x- AIn(x)) = In(x)

=A + (x - A) -x + AIn(x) = AIn(x)

Comparing LHS to RHS,we have A=1.

Therefore, replacing A with 1 we have y=x-Inx