Question

Verify that y² = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]

Answer 1

We have, \[y^2 = 4ax ...........(1)\]

Differentiating both sides of (1) with respect to x, we get
\[2y\frac{dy}{dx} = 4a\]
⇒ \[\frac{dy}{dx} = \frac{2a}{y}  ...........(2)\]

Now, differentiating both sides of (1) with respect to y, we get
\[2y = 4a\frac{dx}{dy}\]
⇒ \[\frac{dx}{dy} = \frac{y}{2a}..............(3)\]

\[\therefore x\frac{dy}{dx} + a\frac{dx}{dy} = x\left( \frac{2a}{y} \right) + a\left( \frac{y}{2a} \right) ..........\left[\text{Using  (2) and (3)}\right]\]
\[ \Rightarrow x\frac{dy}{dx} + a\frac{dx}{dy} = \frac{2ax}{y} + \frac{y}{2}\]
\[ \Rightarrow x\frac{dy}{dx} + a\frac{dx}{dy} = \frac{y^2}{2y} + \frac{y}{2} ..........\left[\text{Using (1)}\right]\]
\[ \Rightarrow x\frac{dy}{dx} + a\frac{dx}{dy} = \frac{y}{2} + \frac{y}{2}\]
\[ \Rightarrow x\frac{dy}{dx} + a\frac{dx}{dy} = y\]
\[\Rightarrow y = x\frac{dy}{dx} + a\frac{dx}{dy}\]

Hence, the given function is the solution to the given differential equation.