Differential Equations, Homogeneous Differential Equations, Solution of Homogeneous Differential Equation

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Homogeneous Differential Equations:

Consider the following differential equations:

In equation (i) above, we see that each term except is of degree

[As degree of is , degree of is and degree of is ]

In equation (ii) each term except is of degree.

In equation (iii) each term except is of degree, as it can be rewritten as

Such equations are called homogeneous equations.


Homogeneous equations do not have constant terms.

Solution of Homogeneous Differential Equation:

To solve such equations, we proceed in the following manner:

I. Write one variable. (the other variable).

(i.e. either or )

II. Reduce the equation to separable form

III. Solve the equation as we had done earlier.




The given differential equation is


It is a homogeneous equation of degree two.

Let . Then

From (1), we have


Or or

Or or


Further on integrating both sides of (2), we get

where is an arbitrary constant.

On substituting the value of , we get

which is the required solution.


If the Homogeneous differential equation is written in the form then is substituted to find solution

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