Question

Verify that y = cx + 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,
$y=cx+2c^2$ ...(1)
Differentiating both sides of (1) with respect to $x$, we get
$\frac{dy}{dx}=c$ ...(2)
Now,
$$\begin{gathered} 2{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y \\ =2c^2+cx-cx-2c^2=0\left[\mathrm{Using}\left(1\right)\mathrm{and}\left(2\right)\right] \\ \Rightarrow 2{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y=0 \end{gathered}$$
Hence, the given function is the solution to the given differential equation.