Question

please, help Find the general solution of the differential equation
(4x-y)dx=xdy

Answer 1

We have to find the general solution of (4x - y)dx = x dy

solution : (4x - y) dx = x dy

⇒(4x - y)/x = dy/dx

⇒4x/x - y/x = dy/dx

⇒4 - (y/x) = dy/dx

Now let y/x = P.....(1)

⇒y = Px

differentiating with respect to x,

⇒dy/dx = xdP/dx + P .....(2)

Now putting equations (1) and (2) in differential equation we get,

4 - P = x dP/dx + P

⇒4 - 2P = x dP/dx

⇒∫dP/(4 - 2P) = ∫dx/x

⇒-1/2 log(2 - P) = logx + C

Now putting P = y/x

⇒+1/2 log(2 - y/x) = logx + C

⇒0 = logx + 1/2 log(2 - y/x) + C

⇒ logx√(2 -y/x) = - C

⇒√(2x² - xy) = e^-C

⇒2x² - xy = (e^-C)² = K

⇒2x² - xy = K

Therefore the general solution of given differential is 2x² - xy = k