Method of solving first order Homogeneous differential equation

**Homogeneous Differential Equations**

A differential equation of the form *dy*/dx = *f* (*x*, *y*)/*
g *(*x*,* y*) is called homogeneous differential equation if *f*
(*x*, *y*) and *g*(*x*, *y*) are homogeneous functions
of the same degree in *x* and *y*. (or)

Homogeneous differential can be written as dy/dx = F(y/x).

Check *f* (
*x*, *y*)
and *g* ( *x*, *y*) are homogeneous
functions of same degree.

The given differential equation becomes v x dv/dx =F(*v*)

Separating the variables, we get

By integrating we get the solution in terms of *v* and *x*.

Replacing
v by y/x we get the solution.

**Example 4.15**

Solve the differential equation *y*^{2} *dx* + ( *xy* + *x*^{2} )*dy*
= 0

*Solution*

**Example 4.17**

Find the particular solution of the differential equation *x*
^{2} *dy* +
*y* ( *x* + *y*) *dx* = 0 given that *x* = 1, *y* = 1

*Solution:*

**Example 4.18**

If the marginal cost of producing *x* shoes is given by (3*xy*
+ *y*^{2} ) *dx*
+ (*x* ^{2} + *xy*) *dy* = 0 and the total cost of
producing a pair of shoes is given by ₹12. Then find the total cost function.

*Solution:*

Given marginal cost function is (x^{2} + xy) dy + (3xy + y^{2})dx=0

**Example 4.19**

The marginal revenue ‘y’ of output ‘q’ is given by the equation . Find the total Revenue function when output is 1 unit and Revenue is ₹5.

*Solution:*

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12th Business Maths and Statistics : Chapter 4 : Differential Equations : Homogeneous Differential Equations | Example Solved Problems with Answer, Solution, Formula

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