The differential equation obtained by eliminating the arbitrary constants from the
equation \(y^{2}=(2 x+c)^{5}\) is

We have \(y^{2}=(2 x+c)^{5}\) ...(i)
\(\therefore 2 \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}=5(2 \mathrm{x}+\mathrm{c})^{4}(2) \Rightarrow \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}=5(2 \mathrm{x}+\mathrm{c})^{4}\)
\(\therefore(2 x+c)=\left[\frac{y}{5}\left(\frac{d y}{d x}\right)\right]^{\frac{1}{4}}\) and substituting value of \((2 x+c)\) in eq. (i), we write
\(y^{2}=\left[\frac{y}{5}\left(\frac{d y}{d x}\right)\right]^{\frac{5}{4}}\)
Raising both sides to power of 4 , we get
\(\begin{aligned}
& y^{8}=\left[\frac{y}{5}\left(\frac{d y}{d x}\right)\right]^{5}=\frac{y^{5}}{3125}\left(\frac{d y}{d x}\right)^{5} \\
\therefore & 3125 y^{3}=\left(\frac{d y}{d x}\right)^{5}
\end{aligned}\)