Final answer:
To find the general solution of the given differential equation x dy/dx + 2y = x³-x, we can solve it using the integrating factor method. The general solution is y = (x-1/3) + C/x², where C is an arbitrary constant.
Step-by-step explanation:
To find the general solution of the given differential equation x dy/dx + 2y = x³-x, we can write it in the standard form dy/dx + (2/x)y = (x²-1)/x.
This is a first-order linear homogeneous differential equation, and we can solve it using the integrating factor method.
- First, we find the integrating factor I(x) = e^(∫(2/x)dx) = e^(2ln|x|) = x².
- Multiplying both sides of the equation by the integrating factor, we get x²(dy/dx) + 2xy = (x²-1).
- Integrating both sides with respect to x, we have ∫(x²(dy/dx) + 2xy)dx = ∫(x²-1)dx.
- Simplifying and integrating, we get x³y + x²y' + Cx = x³/3 - x + C.
- Finally, rearranging the terms gives the general solution y = (x-1/3) + C/x², where C is an arbitrary constant.