Question

t%5E%7B2%7D%20%7D%20" id="TexFormula1" title=" \frac{dx}{dt} = \frac{t^{2}+3tx+ x^{2} }{ t^{2} } " alt=" \frac{dx}{dt} = \frac{t^{2}+3tx+ x^{2} }{ t^{2} } " align="absmiddle" class="latex-formula">

Answer 1

$\dfrac{\mathrm dx}{\mathrm dt}=\dfrac{t^2+3tx+x^2}{t^2}$
$\dfrac{\mathrm dx}{\mathrm dt}=1+\dfrac{3x}t+\dfrac{x^2}{t^2}$

Let $x(t)=ty(t)$, so that $\dfrac{\mathrm dx}{\mathrm dt}=t\dfrac{\mathrm dy}{\mathrm dt}+y$.

$t\dfrac{\mathrm dy}{\mathrm dt}+y=1+3y+y^2$
$t\dfrac{\mathrm dy}{\mathrm dt}=1+2y+y^2=(1+y)^2$
$\dfrac{\mathrm dy}{(1+y)^2}=\dfrac{\mathrm dt}t$
$\implies-\dfrac1{1+y}=\ln|t|+C$
$\implies y+1=-\dfrac1{\ln|t|+C}$
$\implies yt=x=-t-\dfrac t{\ln|t|+C}$