## 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 \(\frac { dy }{ dx } =\frac { f(x,y) }{ g(x,y) }\) 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, \(\int { \frac { 1 }{ F(v)-v } } dv\quad =\quad \int { \frac { dx }{ x } +c }\) where c is an arbitrary constant of integration. After integration, v will be replaced by \(\frac { y }{ x }\) 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

**1.**

**Solution:**

**2.**

**Solution:**

**3.**

**Solution:**

**4.**

**Solution:**

**5.**

**Solution:**

**6.**

**Solution:**

**7.**

**Solution:**

**8.**

**Solution:**

**9.**

**Solution:**

**10.**

**Solution:**

**11.**

**Solution:**

**12.**

**Solution:**