Let y=y(x) solve y^2,dx+(x^2-xy+y^2),dy=0, passing through (1,1), and intersects y=√3x at …

Question

Let $y=y(x)$ solve $y^2\,dx+(x^2-xy+y^2)\,dy=0$, passing through $(1,1)$, and intersects $y=\sqrt3x$ at $(\alpha,\sqrt3\alpha)$. Then $\log_e(\sqrt3\alpha)$ equals:

Accepted Answer

With $y=vx$, the homogeneous ODE reduces and integrating with $(1,1)$ gives a relation; substituting the intersection with $y=\sqrt3x$ (i.e. $v=\sqrt3$) yields $\log_e(\sqrt3\alpha)=\dfrac{\pi}{12}$.

Solutions · 1