How do you find all solutions of the differential equation (d^3y)/(dx^3)=e^x?

Jan 10, 2017

$y = {e}^{x} + A {x}^{2} + B x + C$

Explanation:

This is a third order separable differential equation which we can solver by repeated integration, (or separating the variables):

$\frac{{d}^{3} y}{{\mathrm{dx}}^{3}} = {e}^{x}$

Integrating we get

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

And a second time:

$\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{x} + {A}_{1} x + B$

And a third time:

$y = {e}^{x} + {A}_{1} {x}^{2} / 2 + B x + C$

So we can write the GS as;

$y = {e}^{x} + A {x}^{2} + B x + C$