Ch 05/Homogeneous Equations

Homogeneous Equations

A DE is homogeneous if f(x,y) depends only on y/x. Substitute v = y/x to convert to separable.

IThe Homogeneous Substitution

A DE is homogeneous if for all t — equivalently, if f can be written as a function of alone.

Substitution: Let , so and differentiating:

Substitute these into the DE. The result is always a separable DE in v and x. Solve for v, then back-substitute .

Note

How to check if a DE is homogeneous: try to express purely as a function of . If you can, substitute .

Example 1medium

Solve .

Show step-by-step solution+

Verify homogeneous

Divide top and bottom by : — yes, it's a function of

Substitute y = vx, dy/dx = v + xv'

Isolate xv'

Separate

Integrate both sides

and

Combine

Back-substitute v = y/x

, so

General solution

Example 2easy

Solve .

Show step-by-step solution+

Write as dy/dx

— function of

Substitute y = vx

Simplify

Separate

Integrate

Back-substitute v = y/x

Example 3hard

Solve .

Show step-by-step solution+

Write as dy/dx

Divide by x to see v = y/x form

Substitute

Simplify xv'

Separate

Split left side

Integrate

Back-substitute v = y/x

← Previous

Ch 4 · Variable Separable Method

Next chapter →

Ch 6 · Linear DE — Integrating Factor Method