Question

Find the equation of the curve which passes through the point (3, −4) and has the slope $\frac{2y}{x}$ at any point (x, y) on it.

Answer 1

According to the question,
$\frac{dy}{dx}=\frac{2y}{x}$
$$\begin{gathered} \Rightarrow \frac{1}{2y}dy=\frac{1}{x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ \int \frac{1}{2y}dy=\int \frac{1}{x}dx \\ \Rightarrow \frac{1}{2}\log \left|y\right|=\log \left|x\right|+C \\ \mathrm{Since}\mathrm{the}\mathrm{curve}\mathrm{passes}\mathrm{through}\left(3,-4\right),\mathrm{it}\mathrm{satisfies}\mathrm{the}\mathrm{above}\mathrm{equation}. \\ \therefore \frac{1}{2}\log \left|-4\right|=\log \left|3\right|+C \\ \Rightarrow \log \left|2\right|-\log \left|3\right|=C \\ \Rightarrow C=\log \left|\frac{2}{3}\right| \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}C,\mathrm{we}\mathrm{get} \\ \log \left|y\right|=2\log \left|x\right|+2\log \left|\frac{2}{3}\right| \\ \Rightarrow \log \left|y\right|=\log \left|\frac{4}{9}{x}^2\right| \\ \Rightarrow y=\pm \frac{4}{9}{x}^2 \\ \Rightarrow 9y-4x^2=0or9y+4x^2=0 \\ \mathrm{The}\mathrm{given}\mathrm{point}\mathrm{does}\mathrm{not}\mathrm{satisfy}\mathrm{the}\mathrm{equation}9y-4x^2=0. \\ \therefore 9y+4x^2=0 \end{gathered}$$