Homogeneous equations

Some equations cannot be separated directly but become separable with a clever substitution. These are homogeneous equations.

Test: F(x, y) is homogeneous of degree n if replacing x by λx and y by λy gives λⁿ F(x, y). For example, x² + xy is homogeneous of degree 2, while x² + y is not homogeneous.

dy/dx = F(x, y) is a homogeneous differential equation when F has degree 0, which means F can be written using only y/x. Quick check: every term on the top and on the bottom has the same total power.

Method:

• Put y = vx. Then dy/dx = v + x dv/dx (product rule).
• Substitute: the x's cancel, leaving an equation in v and x only.
• Separate v and x, and integrate.
• Finally put v = y/x back.

Example: dy/dx = (x² + y²)/(2xy).

• Put y = vx: v + x dv/dx = (1 + v²)/(2v).
• So x dv/dx = (1 – v²)/(2v), and 2v dv/(1 – v²) = dx/x.
• Integrate: –log|1 – v²| = log|x| + c, so x(1 – v²) = A.
• Put v = y/x: x² – y² = Ax.

If the equation is written as dx/dy and depends on x/y, put x = vy instead, with dx/dy = v + y dv/dy.