`dy/dx = 3x^2/y^2` Find the general solution of the differential equation

Friday, April 29, 2016

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}={3}\frac{{{x}}^{{2}}}{{{y}}^{{2}}}$

This differential equation is separable since it can be rewritten in the form

$\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$

So separating the variables, the equation becomes

$\displaystyle{{y}}^{{2}}{\left.{d}{y}\right.}={3}{{x}}^{{2}}{\left.{d}{x}\right.}$

Taking the integral of both sides, the equation becomes

$\displaystyle\int{{y}}^{{2}}{\left.{d}{y}\right.}=\int{3}{{x}}^{{2}}{\left.{d}{x}\right.}$

$\displaystyle\frac{{{y}}^{{3}}}{{3}}+{C}_{{1}}={3}\cdot\frac{{{x}}^{{3}}}{{3}}+{C}_{{2}}$

$\displaystyle\frac{{{y}}^{{3}}}{{3}}+{C}_{{1}}={{x}}^{{3}}+{C}_{{2}}$

Since C1 and C2 represent any number, it can be expressed as a single constant C.

$\displaystyle\frac{{{y}}^{{3}}}{{3}}={{x}}^{{3}}+{C}$

Therefore, the general solution of the given $\displaystyle\frac{{{y}}^{{3}}}{{3}}={{x}}^{{3}}+{C}$ .

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