The general solution of the differential equation [(x-y^2) d x+y(5 x+y^2) d y=0] is :

Question

The general solution of the differential equation $$\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0$$ is :

Accepted Answer

$\left(x-y^{2}\right) d x+y\left(5 x+y^{2}\right) d y=0$

$$
y \frac{d y}{d x}=\frac{y^{2}-x}{5 x+y^{2}}
$$

Let $y^{2}=t$

$$
\frac{1}{2} \cdot \frac{d t}{d x}=\frac{t-x}{5 x+t}
$$

Now substitute, $t=v x$

$$
\begin{aligned}
& \frac{d t}{d x}=v+x \frac{d v}{d x} \\
& \frac{1}{2}\left\{v+x \frac{d v}{d x}\right\}=\frac{v-1}{5+v} \\
& x \frac{d v}{d x}=\frac{2 v-2}{5+v}-v=\frac{-3 v-v^{2}-2}{5+v} \\
& \int \frac{5+v}{v^{2}+3 v+2} d v=\int-\frac{d x}{x} \\
& \int \frac{4}{v+1} d v-\int \frac{3}{v+2} d v=-\int \frac{d x}{x} \\
& 4 \ln |v+1|-3 \ln |v+2|=-\ln x+\ln C \\
& \left|\frac{(v+1)^{4}}{(v+2)^{3}}\right|=\frac{c}{x} \\
& \left| \frac{\left(\frac{y^{2}}{x}+1\right)^{4}}{\left(\frac{y^{2}}{x}+2\right)^{3}}\right|=\frac{c}{x} \\
& \left|\left(y^{2}+x\right)^{4}\right|=C\left|\left(y^{2}+2 x\right)^{3}\right|
\end{aligned}
$$

Solutions · 1