Solve $(t^2+3ty+y^2)-t^2y'=0, y(1)=0$, for $t>0$

Question

$(t^2+3ty+y^2)-t^2y'=0, y(1)=0$, for $t>0$.
$$\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial t}}{N}=\frac{5t+2y}{-t^2}$$which is not a function of t only. Then I am stuck. Could anyone help? Thanks

Answer 1

It's homogeneous

Substitute $y=tx \implies y'=x't+x$ $$(t^2+3ty+y^2)-t^2y'=0, y(1)=0$$ $$(t^2+3t^2x+t^2x^2)-t^2(x't+x)=0$$ $$tx'+x=x^2+3x+1$$ $$tx'=(x+1)^2$$ this equation is separable

Answer 2

Note that the equation becomes:

$$y' = \frac{t^2 + 3ty + y^2}{t^2}$$

which is a homogeneous ODE. Now substitute $y=ut$ to reduce it to an equation which can be easily solved.