Question

The equation of the curve whose slope is given by $\frac{dy}{dx}=\frac{2y}{x};x>0,y>0$ and which passes through the point (1, 1) is
(a) x² = y
(b) y² = x
(c) x² = 2y
(d) y² = 2x

Answer 1

(a) x² = y

We have,
$$\begin{gathered} \frac{dy}{dx}=\frac{2y}{x} \\ \Rightarrow \frac{1}{2}\times \frac{1}{y}dy=\frac{1}{x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \frac{1}{2}\int \frac{1}{y}dy=\int \frac{1}{x}dx \\ \Rightarrow \frac{1}{2}\log y=\log x+\log C \\ \Rightarrow \log y^{\frac{1}{2}}-\log x=\log C \\ \Rightarrow \log \left(\frac{\sqrt{y}}{x}\right)=\log C \\ \Rightarrow \frac{\sqrt{y}}{x}=C \\ \Rightarrow \sqrt{y}=Cx.....\left(1\right) \\ \mathrm{As}\left(1\right)\mathrm{passes}\mathrm{through}(1,1),\mathrm{we}\mathrm{get} \\ \therefore 1=C \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}C\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \sqrt{y}=x \\ \Rightarrow y=x^2 \end{gathered}$$