Solving a Nonlinear Differential Equation with Multiple Methods

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Homework Statement



Solve the follwing differential equation
2y'-(x/y)+x^3 cosy = 0 solve?

Homework Equations



Linear Differential equation: y'+py=q
exact differential equation: Mdx+Ndy=0

The Attempt at a Solution



both of the methods i have applied ... but didnt work
later i tried it as Bernoulli Differential equation in x... but variables are not coming in

x' + px = q form...


kindly help me or give me hint to solve the above differential equation.
 
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This is a non-linear ODE because of cos y. Standard linear ODE solution techniques won't work.
 
Give me some hint ... I have also tried Cauchy and legendre methods as well ... But no one works here
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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