to v + x dv / dx = F ( v )
=> x dv / dx = F ( v ) – v ∴ dv / F ( v ) – v = dx / x
Hence , the variables are separated in terms of v and x .
5 . Differential Equations Reducible to Homogeneous Equation
The differential equation of the form
dy / dx = a1x + b1y + c1 / a2x + b2y + c2 ……( i ) put X = X + h and y = Y + k ∴ dY / dX = a1 X + b1 Y + ( a1h + b1k + c1 ) / a2X + b2 Y + ( a2h + b2k + c2 ) ……( ii )
We choose h and k , so as to satisfy a1h + b1k + c1 = 0 and a2h + b2k + c2 = 0 . On solving , we get
h / b1c2 – b2c1 = k / c1a2 – c2a1 = 1 / a1b2 – a2b1 ∴ h = b1c2 – b2c1 / a1b2 – a2b1 and k = c1a2 – c2a1 / a1b2 – a2b1
Provided a1b2 – a2b1 ≠ 0 , a1 / a2 ≠ ba / b2
Then , Eq , ( ii ) reduces to dY / dX = ( a1 X + b1 Y ) / ( a2X + b2 Y ), which is a homogeneous form and will be solved easily .