Question

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

Answer 1

`(\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`