# Find a way to solve the following differential equation!

I am not a mathematician so I apologize in advance if my post represents an easy problem.

Suppose that I have the following differential equation

$$\eta(x)''+\omega^2 \eta(x)=\omega^2 R$$,

where $$\omega^2>0$$ and $$R>0$$ are known physical parameters and let's say $$\eta$$ is the unknown function that depends on $$x$$. This is easy to solve and the general solution reads

$$\eta(x)=\alpha_1 \sin(\omega x)+\alpha_2 \cos(\omega x)+R$$,

with $$\alpha_1$$ and $$\alpha_2$$ as the constants that should be calculated upon applying the boundary conditions.

Now suppose that I make my differential equation harder by changing $$\eta(x)$$ to a more involved expression, i.e.

$$\eta(x)''+\omega^2 \big( \sqrt{(\eta(x)-\eta^*)^2} -\eta^* \big)=\omega^2 R$$,

where $$\eta^*>0$$ is an additional physical parameter, which is known.

However, I have no idea how to solve this differential equation?

My first question is how to solve this differential equation?

Further, assume that I substitute the expression in the parentheses with another expression, e.g. a $$\tan$$ function. Now the second question is: is there any general way to solve the differential equation for any function in the parentheses?

Just let $$y(x) = \eta(x) - \eta^*$$. As $$\eta^*$$ is a constant, we get $$y''(x) = \eta''(x)$$.

So, our differential equation becomes - $$y''(x)+\omega^2|y(x)| = \omega^2 (R+\eta^*)$$

This is almost the same equation as before, but with a $$|y|$$ instead of $$y$$.

Let us solve this case by case -

1. Let $$y\ge 0$$. Then, we have $$y''(x)+\omega^2y(x) = \omega^2 (R+\eta^*)$$, with the solution $$y(x)=\alpha_1 \sin(\omega x)+\alpha_2 \cos(\omega x)+R+\eta^*$$, valid only when $$y\ge 0$$

2. Let $$y\le 0$$. Then, we have $$y''(x)-\omega^2y(x) = \omega^2 (R+\eta^*)$$, with the solution $$y(x) = \beta_1 e^{\omega x} + \beta_2 e^{-\omega x} - R -\eta^*$$, valid only when $$y\le0$$

About the $$2^{nd}$$ question, it is in general not possible to get a closed form solution for $$\eta(x)$$. Take as an example - $$\eta''(x) + sin(\eta(x)) = 0$$.

However, we can say something about whether a solution exists. If it exists, we can use numerical methods to get to a solution. For example, let us have $$\eta''(x) + f(\eta(x))$$ as our general ODE. Then, one result (the Peano existence theorem) says that if $$f(x)$$ is continuous, then a solution of the ODE exists (Note that the solution need not be unique)

• Thanks for your response. Could you also comment on my second question? – KratosMath Nov 6 '19 at 19:03
• Sorry, forgot to see that! – Ishan Deo Nov 6 '19 at 19:20