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(dx)/(dt) = (t^(2)+3tx+ x^(2) )/( t^(2) )

User Xiy
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(\mathrm dx)/(\mathrm dt)=(t^2+3tx+x^2)/(t^2)

(\mathrm dx)/(\mathrm dt)=1+\frac{3x}t+(x^2)/(t^2)

Let
x(t)=ty(t), so that
(\mathrm dx)/(\mathrm dt)=t(\mathrm dy)/(\mathrm dt)+y.


t(\mathrm dy)/(\mathrm dt)+y=1+3y+y^2

t(\mathrm dy)/(\mathrm dt)=1+2y+y^2=(1+y)^2

(\mathrm dy)/((1+y)^2)=\frac{\mathrm dt}t

\implies-\frac1{1+y}=\ln|t|+C

\implies y+1=-\frac1+C

\implies yt=x=-t-\frac t\ln
User Mitzi
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