 # Solve the first-order differential equation State which method you are using! x^2y^2dx-(x^3+1)dy=0 prsategazd 2022-01-20 Answered
Solve the first-order differential equation State which method you are using!
${x}^{2}{y}^{2}dx-\left({x}^{3}+1\right)dy=0$
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${x}^{2}{y}^{2}dx-\left({x}^{3}+1\right)dy=0$
$\left({x}^{3}+1\right)dy={x}^{2}{y}^{2}dx$

$\frac{dy}{{y}^{2}}=\left(\frac{{x}^{2}}{{x}^{3}+1}\right)dx$
Variable separable for
$\int \frac{1}{{y}^{2}}dy=\int \frac{{x}^{2}}{{x}^{3}+1}dx$
$\int \frac{1}{{y}^{2}}dy=\mid \frac{1}{3}\int \frac{3{x}^{2}}{{x}^{3}+1}dx$
$\frac{-1}{y}=\frac{1}{3}\mathrm{ln}\left({x}^{3}+1\right)+c$
$\frac{1}{y}=\frac{-1}{3}\mathrm{ln}\left({x}^{3}+1\right)+{c}_{1}$
or
$y=\frac{-3}{\mathrm{ln}\left({x}^{3}+1\right)}+{c}_{2}$

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