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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?

Thanks in advance,

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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)

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

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