The differential equation obtained by eliminating arbitrary constant from the equation \(y^2=(x+c)^3\) is

\(y^2=(x+c)^3\)
Differentiating w.r.to \(x\), we get
\(\begin{aligned}
& 2 y \frac{\mathrm{~d} y}{\mathrm{~d} x}=3(x+\mathrm{c})^2 \\
& \Rightarrow(x+\mathrm{c})^2=\frac{2 y}{3} \frac{\mathrm{~d} y}{\mathrm{~d} x} \\
& \Rightarrow(x+\mathrm{c})^6=\left(\frac{2 y}{3} \frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3 \\
& \Rightarrow\left((x+\mathrm{c})^3\right)^2=\frac{8 y^3}{27}\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3 \\
& \Rightarrow\left(y^2\right)^2=\frac{8 y^3}{27}\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3 \\
& \Rightarrow y^4=\frac{8 y^3}{27}\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3 \\
& \Rightarrow 27 y=8\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3
\end{aligned}\)