The solution curve, of the differential equation [2 y d y~d x+3=5 d y~d x], passing throug…

Question

The solution curve, of the differential equation $$2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}$$, passing through the point $(0,1)$ is a conic, whose vertex lies on the line :

Accepted Answer

$$\begin{aligned}
& 2 y \frac{d y}{d x}+3=5 \frac{d y}{d x} \
& 2 y d y+3 d x=5 d y \
& y^2+3 x=5 y+\left.c\right|_{(0,1)} \
& 1+0=5+c \
& c=-4 \
& y^2-5 y=-3 x-4 \
& y^2-5 y+\frac{25}{4}-\frac{25}{4}=-3 x-4 \
& \left(y-\frac{5}{2}\right)^2=-3 x+\frac{9}{4} \
& \left(y-\frac{5}{2}\right)^2=-3\left(x-\frac{3}{4}\right) \
& \left(\frac{3}{4}, \frac{5}{2}\right) \text { satisfies by } 2 x+3 y=9
\end{aligned}$$

Solutions · 1