# Difficult Second Order Nonlinear Differential Equation

I've been attempting to create a model for a particular physical phenomena. I've reached the stage where I need to find the solution to a differential equation in order to continue. I've attempted to solve it on my own through a substitution but was unable to get anywhere with it. I am aware that the differential equation may not solvable so I will provide a few different forms. If any of these are solvable please let me know.

a) $$y'' + (y')^2 = sin(y)$$

b) $$y'' + (y')^2 = y$$

Thank you for your time and help.

• Do you have initial or boundary conditions? Are you considering numerical solutions or only analytical solutions? Commented Dec 23, 2018 at 19:01
• I had an analytical solution in mind. However, if this is not possible a numerical solution would suffice. Commented Dec 23, 2018 at 19:05
• Note that $y''=\dfrac{\mathrm{d}}{\mathrm{d}y}(y'^2/2)$ to reduce to a first-order linear equation in $y'^2$... Commented Dec 23, 2018 at 19:22
• If you just want the general behavior of the solution around a point $x_0$, one may try to do local analysis such as dominated balance to obtain an approximate solution. Commented Dec 23, 2018 at 21:08

Let $$u = (y')^2$$. You can transform as follows:

$$y'' = \frac{\frac{d}{dx}(y'^2)}{2y'} = \frac12\frac{\dfrac{du}{dx}}{\dfrac{dy}{dx}} = \frac12\frac{du}{dy}$$

Then

$$\frac{du}{dy} + 2u = 2g(y)$$

where $$g(y)$$ is any function on your RHS. This is a standard first-order equation in $$u(y)$$.

Once you have $$u(y)$$, it remains to solve

$$\frac{dy}{dx} = \pm\sqrt{u(y)}$$

which is a separable equation

• It would appear that when $g(y) = sin(y)$ or $g(y) = y$, it is not possible to integrate $u(y)$. Commented Dec 24, 2018 at 9:56
• It usually isn't possible to represent the solution in closed form. If you have some initial conditions, you can use numerical integration. Commented Dec 24, 2018 at 10:05