Question

Q1 Verify that y2 =4ax is a solution of the differential equation y= x dy/dx + a dx/dy

Answer 1

$\textbf{Given:}$

$y^2 =4ax$............(1)

$\text{Differentiate (1) with respect to x}$

$2y\;\frac{dy}{dx}=4a$

$\implies\bf\frac{dy}{dx}=\frac{2a}{y}$

$\text{Differentiate (1) with respect to y}$

$2y=4a\;\frac{dx}{dy}$

$\implies\bf\frac{dx}{dy}=\frac{y}{2a}$

$\text{Now,}$

$x\,\frac{dy}{dx}+a\,\frac{dx}{dy}$

$=x(\frac{2a}{y})+a(\frac{y}{2a})$

$=\frac{2ax}{y}+\frac{y}{2}$

$=\frac{\frac{y^2}{2}}{y}+\frac{y}{2}$

$=\frac{y}{2}+\frac{y}{2}$

$=y$

$\implies\boxed{\bf\,x\,\frac{dy}{dx}+a\,\frac{dx}{dy}=y}$