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.