Question

Verify that the function y=cx+a/c is the general solution of the differential equation, y=x(dy/dx)+a(dx/dy) (c is an arbitrary constant.)

Answer 1

Answer:

Given:

$\rm y = cx+\dfrac ac.$

where, c is an arbitrary constant.

To verify:

$\rm y=x\left( \dfrac{dy}{dx}\right )+a\left( \dfrac{dx}{dy}\right ).$

Now,

$\rm \dfrac {dy}{dx}=\dfrac {d}{dx}\left(cx+\dfrac ac\right )=c+0=c.\\\therefore \dfrac {dx}{dy}=\dfrac{1}{\dfrac {dy}{dx}}=\dfrac 1c.$

On putting both these values in the RHS of the given differential equation,

$\rm RHS=x\left( \dfrac{dy}{dx}\right )+a\left( \dfrac{dx}{dy}\right )=xc+\dfrac ac.$

and LHS of the equation is given by

$\rm LHS = y=cx+\dfrac ac.$

Clearly, LHS = RHS, thus the given expression is verified.