# Is this differential equation solvable? $\frac{1}{2} \left( \frac{dV}{dx} \right)^2 = a + b x + c \exp(\alpha x)$

I'm trying to test the output of a numerical simulation and want to come up with an analytic expression to test it. The differential equation I came up with is

$$\frac{1}{2} \left( \frac{dV}{dx} \right)^2 = a + b x + c \exp(\alpha x)$$

And I cannot figure out how to solve it with the square / square root in the expression. Any advice?

• You can solve it analytically for $c=0$ which atleast gives you some test-cases to work with. The result here is $V(x) = V(0) + \sqrt{2} \left[\frac{2}{3b}(a + bx)^{3/2} - a^{3/2}\right]$ (assuming $V'>0$ otherwise there is a minus sign in from of the second term). – Winther Nov 29 '16 at 15:08
• For $b=0$ too ! – Claude Leibovici Nov 29 '16 at 15:09

Simply $$V=\int_0^x\sqrt{2(a+bx+c\exp(\alpha x))}dx,$$ which has no closed-form expression.