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