Final answer:
The differential equation x³y' = x´y - 3x´ can be solved by first rearranging it, finding an integrating factor, and then integrating to obtain the general solution.
Step-by-step explanation:
To find the general solution of the differential equation x³y' = x´y - 3x´, we start by arranging it into a standard form for solving. This can be written as y' - xy = -3 after dividing through by x³, assuming x ≠ 0. Notably, this is a first-order linear differential equation for which the solution involves finding an integrating factor.
An integrating factor μ(x) is generally given by e^{∫ P(x) dx}, where P(x) is the coefficient of y in the standard form. Once that is calculated, we multiply through by the integrating factor, integrate the result, and then solve for y, thereby obtaining the general solution.
Here, P(x) is just -x, so the integrating factor will be e^{-∫ x dx} which simplifies to e^{-x²/2}. After finding the integrating factor, apply it to the differential equation, and integrate to find the general solution.