Solving a Non-Linear ODE: What Method Should I Use?

lkh1986
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Homework Statement



Solve y'=x^2+y^2 with initial condition y(0)=1.


Homework Equations


This is a first order ODE.



The Attempt at a Solution


I have tried separable variable, exact, and homogeneous and non-homogeneous, but none of them work. It's neither linear nor Bernoulli.

Any clue on what method have I missed or should I tried? Thanks.
 
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It is a http://en.wikipedia.org/wiki/Riccati_equation" .
 
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Metaleer said:
It is a http://en.wikipedia.org/wiki/Riccati_equation" .

Thanks. I will try using some suitable transformation to reduce it to a solvable linear DE. :)
 
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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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