A differential equation of first order and first degree invokes x,y and So it can be put in any one of the following forms :

where f(x,y) and g(x,y) are obviously the function of x,y

**Geometrical Interpretation of the differential equations of first order and first degree**

The general form of a first order and first degree differential equation is

We know that the tangent of the direction of a curve in Cartesian rectangular coordinates at any point given by so the equation in (i) cann be known as an equation which establishes the relationship between the coordinates of a point and the slope of the tangent i.e., to the integral curve at that point.

Solving the differential equation given by (i) means finding those curves for which the direction of tangent at each point coincides with the direction of the field. All the curves represented by the general solution when taken together will give locus of the differential equation. Since there is one arbitrary constant in the general solution of the equation of first order, the locus of the equation can be said to be made up of single infinity of curves.

**Solution of First order and first degree differential equation**

As discussed earlier a first order and first degree differential equation can written as

where f(x,y) and g(x,y) are obviously the functions of x and y.

It is not always possible to solve this type of equations. This solution of this type of differential equations. The solution of some standard forms and methods of obtaining their solutions.

**Methods of solving a first order first degree differential equation**

In this section we shall discuss several techniques of obtaining solutions of various types of differential equations.

**Type I ** Differential equations of the type

To solve this type of differential equations we integrate both sides to obtain the general solution as discussed below:

Integrating both sides, we obtain

### Example

**Solve **

**Solution:**

Integrating both sides, we get

This is the required solution.

**Type II ** Differential equations of the type

To solve this type of differential equations we integrate both sides to obtain the general solution as discussed under:

Integrating both sides, we obtain

### Example

**Solve**

**Solution:**

Integrating both sides we obtain

This is the required solution.

**Type III ** Equation in variable separable form

If the differential equation can be put in the form f(x) dx=g(y) dy we say that the variable separable and such equation can be solved by integrating on both sides the equation is given by

where C is an arbitrary constant.

There is no need of introducing arbitrary constants on both sides as they can be combined together to give just one arbitrary constant.

### Example

**Solve**

**Solution:**

Integrating both sides, we get

which is the required solution.

**Type IV ** Equation reducible to variable separable form

Differential equations of the form can be reduced to variable separable form by the substitution ax+by+c=v

### Example

**Solve**

**Solution:**

Given that

Put 4x+y+1=v, so that

So, the given equation becomes

Integrating both sides, we get

which is the required solution

**Type V ** Homogeneous Differential Equation

A function f(x,y) is called a homogeneous function of degree n if

For example is a homogeneous function degree 2 because

A homogeneous function f(x,y) of degree n can always be written as

If a first order first degree differential equation is expressible in the form

Such type of equations can be reduced to variable separable form by the substitution y=vx as explained below:

The differential equation can be written as

If y=vx ,then Substituting these x values in we get,

On integration

Where C is an arbitrary constant of integration.

After integration v will be replaced by v/x to get the complete solution.

**Algorithm for solving homogeneous differential equation **

**Step I** Put the differential equation in the form

**Step II** Put y=vx and in the equation in **Step I** and cancel out x from the right hand side. The equation reduces to form

**Step III** Shift v on RHS and separate the variables v and x.

**Step IV** Integrate both sides to obtain the solution in terms of v and x.

**Step V** Replace v by in the solution obtained in **Step IV** to obtain the solution in terms of x and y

### Example

Solve the differential equation given that y=1 when x=1

**Solution:**

The given differential equation is

Since each of the functions is a homogeneous function of degree 2 therefore equation (i) is a homogeneous equation.

Putting y=vx and in (i) we get

Integrating both sides we get

It is given that y=1, when x=1. Putting x=1 y=1 in eq (ii) we get

Putting in (ii), we get

This is the required equation.