The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal, is

Let$y = f \left(x\right)$ be the curve. The equation of the tangent at $(x, y)$to this curve is 

$Y-y=f^{\mathrm{\prime }}\left(x\right)(X-x)$

Putting $X = 0$ in (i), we get

$Y=y-x{f}^{\mathrm{\prime }}\left(x\right)$

This ordinate is called the initial ordinate of the tangent. It is given that Initial ordinate of the tangent = Subnormal

$\Rightarrow y-x{f}^{\mathrm{\prime }}\left(x\right)=y\frac{dy}{dx}$

$\begin{array}{l}\Rightarrow y-x\frac{dy}{dx}=y\frac{dy}{dx}\left[\because y=f\left(x\right)\therefore \frac{dy}{dx}=f^{\mathrm{\prime }}\left(x\right)\right]\\ \Rightarrow \frac{dy}{dx}=\frac{y}{x+y}\end{array}$

It is a homogeneous differential equation. It can also be written as 

$\frac{dx}{dy}-\frac{x}{y}=1$ which is  a linear differential equation