Let y=y(x) solve (3y^2-5x^2)y,dx+2x(x^2-y^2),dy=0, y(1)=1.

Question

Let $y=y(x)$ solve $(3y^2-5x^2)y\,dx+2x(x^2-y^2)\,dy=0$, $y(1)=1$. Then $|(y(2))^3-12y(2)|$ is:

Accepted Answer

With $y=mx$, the ODE reduces to $\dfrac{dx}{x}=\dfrac{2(m^2-1)}{m(m^2-3)}\,dm$; integrating and using $(1,1)$ to fix the constant gives
$$y(y^2-12)\cdot x^{-3/2}= \text{const}\cdot,\quad\text{evaluated at } x=2: \quad y^3-12y=32\sqrt2.$$

Solutions · 1