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Homogeneous Differential Equations

December 10, 2020 by Prasanna

Homogeneous Differential Equations

Homogeneous differential equation

A function f(x,y) is called a homogeneous function of degree if f(λx, λy) = λn f(x, y).
For example, f(x, y) = x2 – y2 + 3xy is a homogeneous function of degree 2.

A homogenous function of degree n can always be written as
Homogeneous Differential Equations 1

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
Homogeneous Differential Equations 2

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
Homogeneous Differential Equations 3
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.
Homogeneous Differential Equations 4
Solution:
Homogeneous Differential Equations 5
2.
Homogeneous Differential Equations 6
Solution:
Homogeneous Differential Equations 7
3.
Homogeneous Differential Equations 8
Solution:
Homogeneous Differential Equations 9
4.
Homogeneous Differential Equations 10
Solution:
Homogeneous Differential Equations 11
5.
Homogeneous Differential Equations 12
Solution:
Homogeneous Differential Equations 13
6.
Homogeneous Differential Equations 14
Solution:
Homogeneous Differential Equations 15
7.
Homogeneous Differential Equations 16
Solution:
Homogeneous Differential Equations 17
8.
Homogeneous Differential Equations 18
Solution:
Homogeneous Differential Equations 19
9.
Homogeneous Differential Equations 20
Solution:
Homogeneous Differential Equations 21
10.
Homogeneous Differential Equations 22
Solution:
Homogeneous Differential Equations 23
11.
Homogeneous Differential Equations 24
Solution:
Homogeneous Differential Equations 25
12.
Homogeneous Differential Equations 26
Solution:
Homogeneous Differential Equations 27

Filed Under: Mathematics Tagged With: Equation reducible to homogeneous form, Homogeneous differential equation

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