Question

Find the general solution of the differential equation x^2y′′+xy′−y=2x for x>0 by first showing that the function yp=−12x+x ln(x)is a particular solution for the differential equation.

Answer 1

`x^2y''+xy'-y &=2x\\ x^2y''+2xy'-xy'-y &=2x\\ (x^2y')'-(xy)' &=2x\\ (x^2y'-xy)' &=2x\\ x^2y'-xy &=x^2+C_1\\ xy'-y &=x+(C_1)/(x)\\ x^2\left((y)/(x)\right)' &=x+(C_1)/(x)\\ \left((y)/(x)\right)' &=(1)/(x)+(C_1)/(x^3)\\ (y)/(x) &=ln(x)-(C_1)/(2x^2)+C_2\\ y &=xln(x)-(C_1)/(2x)+C_2x`