Question

Show that y = C x + 2C² is a solution of the differential equation $2{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y=0$.

Answer 1

We have,
$2{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y=0.....\left(1\right)$
Now,
y = C x + 2C^{2
$$\begin{gathered} \Rightarrow \frac{dy}{dx}=C \\ \mathrm{Putting}\frac{dy}{dx}=C\mathrm{and}y=Cx+2C^2\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \end{gathered}$$}
$$\begin{gathered} \mathrm{LHS}=2{\left(C\right)}^2+x\left(C\right)-\left(Cx+2C^2\right) \\ =2C^2+xC-xC-2C^2 \\ =0=\mathrm{RHS} \end{gathered}$$
Thus, y = C x + 2C² is the solution of the given differential equation.