# Solve for G.S./P.S. for the following differential equations using the

Solve for G.S./P.S. for the following differential equations using the solutions by a change in variables as suggested by the equations.
$\left({t}^{2}{e}^{t}-4{x}^{2}\right)dt+8txdx=0$
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einfachmoipf
$\left({t}^{2}{e}^{t}-4{x}^{2}\right)dt+8txdx=0$
$M={t}^{2}{e}^{t}-4{x}^{2}$
$N=8tx$
${M}_{x}==8x$
${N}_{t}=8x$
$\frac{{M}_{x}-{N}_{t}}{N}=\frac{-16x}{8tx}=\frac{-2}{t}$
Integrating
${e}^{-2\int \frac{1}{t}dt}={e}^{-2\mathrm{ln}t}=\frac{1}{{t}^{2}}$
$M={e}^{t}=\frac{4{x}^{2}}{{t}^{2}}$
$N=\frac{8x}{t}$
${M}_{x}=-\frac{8x}{{t}^{2}}$
${N}_{1}=\frac{-8x}{{t}^{2}}$

$\int \left({e}^{t}-\frac{4{x}^{2}}{{t}^{2}}\right)dt+\int 0dx=C$