Ch 03/Homogeneous Differential Equations/H.K. Dass 3.7

Homogeneous Differential Equations

A DE dy/dx = f(x,y)/g(x,y) is homogeneous if f and g are homogeneous functions of the same degree. Substitute y = vx to convert it into a separable equation.

IWhat is a Homogeneous DE?

A DE of the form is called homogeneous if both and are homogeneous functions of the same degree.

Quick test: Replace every with and every with . If all the 's cancel out, it is homogeneous.

Solving method: Put $y = vx$ (where is a new function of ).

Then: .

This substitution ALWAYS converts a homogeneous DE into a separable one. After solving, replace .

Note

If makes the algebra ugly, try instead. Choose whichever makes the separation cleaner.

Example 1 (Easy)easy

Solve .

Show step-by-step solution+

Write in dy/dx form

. Degree 2 each. Homogeneous.

Put y = vx

Isolate and separate

, so .

Integrate

Replace v = y/x

Example 2 (Medium)medium

Solve .

Show step-by-step solution+

Put y = vx

, so .

Separate

Integrate and replace v = y/x

Example 3 (Hard -- using x = vy)hard

Solve .

Show step-by-step solution+

Try x = vy

. Put .

Substitute

, so .

Separate and integrate

, giving .

Replace v = x/y

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