Differential Equations, General and Particular Solutions, Techniques of Solving in G. a Differential Equation

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General and Particular Solutions:

  • Finding solution of a differential equation is a reverse process.

  • Here we try to find an equation which gives rise to the given differential equation through the process of differentiations and elimination of constants.

  • The equation so found is called the primitive other solution of the differential equation.

Example:

Show that , where and are arbitrary constants, is a solution of the differential equation:

Solution:

We are given that

.....

Differentiating both sides of , we get

.....

Differentiating again, we get

Substituting the values of and in the given differential equation, we get

Or

Techniques of Solving in G.a Differential Equation:

When Variables are Separable:

(i) Differential equation of the type

Consider the differential equation of the type

Or

On integrating both sides, we get

Where is an arbitrary constant. This is the general solution.

Example:

Solve given that when

Solution:

The given differential equation is or …..

On integrating both sides of , we get

or

Or …..

Where is an arbitrary constant.

Since when , therefore, if we substitute these values in we will get

Now, on putting the value of in , we get

or

Which is the required particular solution.

(ii) Differential equations of the type

Consider the differential equation of the type

or …...

In equation, and have been separated from one another.

Therefore, this equation is also known differential equation with variables separable.

To solve such differential equations, we integrate both sides and add an arbitrary constant on one side.

Example:

Solve

Solution:

The given differential equation

Can be written as (Here variables have been separated)

On integrating both sides of, we get

Or

Where is an arbitrary constant.

This is the required solution.

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