Solving First Order First Degree Differential Equation

The different ways of solving differential equation are a follows:The different ways of solving differential equation are a follows:

• Method of separation of variables

• Homogeneous differential equations

• Linear differential equations

Method of Separation of Variables

Definition 7

A first - order first - degree differential equation is of the form

If the function f turns out to be of the form

f(x, y) = p(x) q(y) ….(2)

(i.e., the product of a function of x with a function of y, then the differential equation (1) is said to have the variable x, y separable.)

In this case the differential equation (1) can be written in the alternative form as

If P(x) is a primitive of p(x), and Q(y) is a primitive of , then (3) implies

Q(y) - P(x) = C (real number) ….(4)

This is expressed in integral notation as

We say that (5) is obtained from (3) by integration. Thus, (5) is a primitive of differential equation (1) with f(x, y) given by (2).

Note:

The word 'primitive' has been used both in reference to a differential equation, and in reference to a function, but there should be no confusion. By a primitive of a given function , we mean a function , such that g'(x) = f(x) for all xR. In other words, y = g(x), (xI) is a primitive of the differential equation y' = f(x), (xI).

A differential equation is said to be of the variables separable type if it can be in rearranged the form f(x)dx=g(y)dy whose solution is obtained by direct integration.

Example:

Solve the differential equation

Suggested answer:

By integration

y = tan^-1x+c

Homogeneous Differential Equation

A differential equation is said to be a homogeneous differential equation. If it is of the form

where f and g are homogenous functions, in x and y of the same degree.

Note:

A function f(x,y) is said to be homogenous of degree n if it can be expressed in the form

 

Definition 8:

The first-order first-degree differential equation

is said to be homogeneous, if f is a homogeneous function of degree zero,

_,

In other words, f instead of depending on x and y separately, depends on . Thus,

Examples:

A homogeneous differential equation is solved by using y = vx and transforming the given differential equation to a variables separable form in x and v.

To solve homogeneous differential equation we use the substitution.

y = vx

The equation reduces to variables separable.

Example:

Solve

Suggested answer:

Let y = vx

v = log x +c

Resubstituting for v, we get

Note:

A differential equation is said to be linear when the dependent variable and its derivatives occur in the first degree and are not multiplied together.

When f₁ to f_n as the functions of x, in general linear differential equations of order n.

First Order Linear differential equations

Definition 9:

A first-order differential equation is said to be linear if, in it, the unknown function y and its derivative y' appear with non-negative integral index not greater than one and not as product yy' either.

Hence, a first-order linear differential equation is of the form:

where p and q are given real-valued function on I. The differential equation

is also a linear differential equation, where y is the independent variable and x is the dependent variable.

Let us define for arbitrary chosen a Î I. The term e^P(x) is called an integrating factor, (I.F.), of the differential equation(1) since

which is easily integrable. Integrating (2), we obtain

, (C : parameter) ...(3)

The relation (3) gives a primitive of the differential equation (1).

A Leibnitz's linear differential equation is a linear differential of first order which is of the form.

P and Q are the functions of x only.

Its solution is given by

where is called the integrating factor of the differential equation.

Note:

Where P and Q are functions of y only is a Leibnitz's linear differential equation which is linear in x and its solution is given by

C is a constant of integration.

Example:

Solve

Suggested answer:

Divide by x

The general solution is

Important points

Degree and order of a differential equation

How to form a differential equation

Types of differential equations and their general solutions.