## Homogeneous Differential Equations

### Homogeneous differential equation

A function f(x,y) is called a homogeneous function of degree if f(λx, λy) = λ^{n }f(x, y).

For example, f(x, y) = x^{2} – y^{2} + 3xy is a homogeneous function of degree 2.

A homogenous function of degree n can always be written as

If a first-order first-degree differential equation is expressible in the form where f(x, y) and g(x, y) are homogeneous functions of the same degree, then it is called a homogeneous differential equation. Such type of equations can be reduced to variable separable form by the substitution y = νx. The given differential equation can be written as

On integration, where c is an arbitrary constant of integration. After integration, v will be replaced by in complete solution.

**Equation reducible to homogeneous form**

A first order, first degree differential equation of the form

This is non-homogeneous.

It can be reduced to homogeneous form by certain substitutions. Put x = X + h, y = Y + k.

Where h and k are constants, which are to be determined.

### Homogeneous Differential Equations Problems with Solutions

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