Homogeneous Differential Equation - Practice Questions & MCQ

A function f(x, y) is said to be a homogeneous function of degree n if it satisfies the property

Consider the following examples

Now, if the function is given as 

We can define homogeneous differential equation as follows : 

This equation can be solved by the substitution y = vx.

If the differential equation is of the form

It can be reduced to a homogeneous differential equation as follows:

Put x = X + h, y = Y + k                         ……. (2)

where X and Y are new variables and h and k are constants yet to be chosen

From (2)

dx = dX, dy = dY

Equation (1), thus reduces to

In order to have equation (3) as a homogeneous differential equation, choose h and k such that the following equations are satisfied :

Now, (3) becomes

which is a homogeneous differential equation and can be solved by putting Y = vX.

Note:

If d/a = e/b (=t say) , the above method does not apply.

In such cases, Equation (1) becomes

which can be solved by putting ax + by = v

Separate the variables and integrate to get the required solution.