Homogeneous First-Order Differential Equations

Solve homogeneous first-order ODEs of the form y' = F(y/x) using the substitution y=vx, with a worked example.

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A first-order ODE is called homogeneous (first-order) if it can be written as

dy/dx = F (y/x).

This is not the same as a “homogeneous linear ODE”.
Here “homogeneous” means the right-hand side depends on the ratio y/x only.

Method: substitute y = vx

Let

v = y/x ⇒ y = vx.
Differentiate y = vx:

dy/dx = v + xdv/dx.
Substitute into the ODE. Since y/x = v, the right-hand side becomes F(v):

v + xdv/dx = F(v).
Rearrange to get a separable ODE for v(x):

xdv/dx = F(v)-v.
Solve for v(x), then substitute back y = vx.

Solve

dy/dx = 1 + y/x.

Let v = y/x, so y = vx and y' = v + xv'. Substitute:

v + xdv/dx = 1 + v ⇒ xdv/dx = 1.

Separate and integrate:

dv = dx/x ⇒ v = ln |x| + C.

Substitute back y = vx:

y = x(ln |x| + C).