Suppose we have the following non-linear differential equation <mrow class="MJX-TeXAtom-ORD">

Alexis Meyer

Alexis Meyer

Answered question

2022-05-16

Suppose we have the following non-linear differential equation
x ¨ + ω 2 ( t ) x 1 x 3 = 0
with x ( t ) being a real function (and ω ( t ) being also time-dependent).
Is there an analytical solution?
If not, is there an analytical solution for some particular form of ω ( t )?
If the answer is again no, what software would be recommendable for numerical solution?

Answer & Explanation

Percyaehyq

Percyaehyq

Beginner2022-05-17Added 18 answers

You can make ω explicit with
ω 2 ( t ) = 1 x 4 ( t ) x ¨ ( t ) x ( t ) ,
and any analytical function x ( t ) can do !
indimiamimactjcf

indimiamimactjcf

Beginner2022-05-18Added 3 answers

This ODE has the interesting name of type Ermakov (Yermakov) equation.
And the solving approach is also tricky.

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