# How do you find all solutions of the differential equation (d^2y)/(dx^2)=2x+1?

Jan 10, 2017

$y = \frac{1}{3} {x}^{3} + \frac{1}{2} {x}^{2} + A x + B$

#### Explanation:

This is a Second Order non-homogeneous DE with constant coefficients, but separable so we can just integrate (twice):

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 2 x + 1$

Integrate:

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{2} + x + A$

Integrate again:

$y = \frac{1}{3} {x}^{3} + \frac{1}{2} {x}^{2} + A x + B$

which is the General Solution;