Solve this diferntiel equation :
$\displaystyle {y}'=y+y^{3}$
$\displaystyle x{{y}'=y+\sqrt{x^{2}+y^{2}}}$
1) $\displaystyle \frac{dy}{dx} = y + y^3 \Rightarrow \int \frac{dy}{y + y^3} = \int \, dx$. Do the left hand integral by using partial fractions.
2) The DE re-arranges into $\displaystyle \frac{dy}{dx} = \frac{y}{x} + \sqrt{1 + \left( \frac{y}{x}\right)^2}$ and the standard technique is to make the substitution $\displaystyle y = xv$.
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