Let y=y(x) solve dydx=yx(1+xy^2(1+_ex)), x0, y(1)=3.

Question

Let $y=y(x)$ solve $\dfrac{dy}{dx}=\dfrac{y}{x}\left(1+xy^2(1+\log_ex)\right)$, $x>0$, $y(1)=3$. Then $\dfrac{y^2(x)}{9}$ is:

Accepted Answer

This is a Bernoulli equation in $y$. Dividing by $y^3$ and setting $t=-1/y^2$ reduces it to a linear ODE:
$$\frac{dt}{dx}+\frac{2t}{x} = 2(1+\log_ex).$$

I.F. $=x^2$. Solving and applying $y(1)=3$ (i.e., $t(1)=-1/9$) yields, after simplification,
$$\frac{y^2}{9} = \frac{x^2}{5-2x^3(2+\log_ex^3)}.$$

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