general solution of high DE "variation of parameters"

Question

find the general solution of the non-homogeneous DE:

x²y'' -3xy'+4y=x², x>0

I first solved the associated homogen. equation using cauchy euler method and found y1 and y2 to be

y1=x², y2=x²ln(x)

then used variation of parameter method by making y'' 's coefficient 1

u1= -ln²(x)/2 +c1

u2= ln(x) +c2

and finally my final solution is:

y=x²(-ln²(x)/2 +c1)+ x²ln(x)(ln(x)+c2)

please let me know if my answer is correct

Answer 1

Nice work! You can combine the x²(lnx)² terms so you only have 3 terms. Take care.