# Thread: Help solving xy" - y' = 3x^2

1. ## Help solving xy" - y' = 3x^2

I am a first time differential equations student so I will try my best.

By observation, I have a second order de with non-constant coefficients. Normally with a question like this, I would first attempt the Cauchy-Euler method (substitute y=x^m), and then use variation of parameters to find the particular solution.

However, in this case the power of x does NOT match the degree of derivation of y (x is power one, y is 2nd derivative) and substituting y=x^m comes out with something that doesn't seem quite right to me.

I thought of dividing through by x and finding and integrating factor, but then realized it is not in the correct form to do so.

If someone could help me to find the general solution I'm fairly sure I could tackle the particular one.

2. Originally Posted by LucasV
I am a first time differential equations student so I will try my best.

By observation, I have a second order de with non-constant coefficients. Normally with a question like this, I would first attempt the Cauchy-Euler method (substitute y=x^m), and then use variation of parameters to find the particular solution.

However, in this case the power of x does NOT match the degree of derivation of y (x is power one, y is 2nd derivative) and substituting y=x^m comes out with something that doesn't seem quite right to me.

I thought of dividing through by x and finding and integrating factor, but then realized it is not in the correct form to do so.

If someone could help me to find the general solution I'm fairly sure I could tackle the particular one.
Divide through by x. Substitute y' = u. Solve for u using integrating factor method. Then get y from u.

3. Thanks! I was at school all day and didn't get a chance to try this until now, but it worked beautifully. I took the first and second derivatives of my result and plugged them into the ode. Low and behold, the result was equal to the right hand side of 3x^2! Thanks again for your help. Solved.