Gabriela Duarte

Answered

2022-01-27

Solving a differential equation with trigonometric functions

How could I approach

$y-A\cdot \mathrm{sin}\left(\frac{{d}^{2}y}{{dx}^{2}}\right)=0$

How could I approach

Answer & Explanation

helsinka04

Expert

2022-01-28Added 11 answers

Hint:

Let s=y′.

Then,

$y=A\mathrm{sin}\frac{ds}{dx}=A\mathrm{sin}\frac{ds}{dy}\frac{dy}{dx}=A\mathrm{sin}s\frac{ds}{dy}$

Rearranging yields

$sds=dy{\mathrm{sin}}^{-1}\frac{y}{A}$

You should finally obtain

$\sqrt{2}x+{C}_{2}=\int dy{(y{\mathrm{sin}}^{-1}\frac{y}{A}+A\sqrt{1-\frac{{y}^{2}}{{A}^{2}}}+{C}_{1})}^{-\frac{1}{2}}$

The integral can be rewritten into

$A\int \frac{\mathrm{cos}g}{\sqrt{{C}_{1}+A(g\mathrm{sin}g+\mathrm{cos}g)}}dg$

by letting$y=A\mathrm{sin}g$

I do not expect the integral is elementary

Let s=y′.

Then,

Rearranging yields

You should finally obtain

The integral can be rewritten into

by letting

I do not expect the integral is elementary

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