# RS Aggarwal Class 12 Homogeneous Integral equations PDF

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### What are Homogeneous Integral equations?

A homogeneous integral equation is a type of integral equation in which the function to be determined and the kernel of the integral equation share the same properties of scaling.

It is an equation of the form

∫ K(x,t)f(t) dt =0

where K(x,t) is the kernel of the integral equation and f(t) is the unknown function to be determined, and the integral is taken over a given interval.

It is called homogeneous because the equation is unchanged when the function f(t) is multiplied by a constant.

To solve a homogeneous integral equation, one typically starts by assuming that the solution is of a certain form, such as a polynomial or a trigonometric function, and then uses that form to find the constant of integration.

There are different methods to solve homogeneous integral equations, such as the method of successive approximations and the method of eigenfunction expansions.

In this method, we use some known function as a trial function and form an integral equation using it. Then we try to find the function that makes the integral equation true.

An example of a Homogeneous Integral equation is:

∫_0^1 (t-x)f(t)dt =0

It can be solved by assuming f(t) = at^n where n is a constant and a is a constant of integration.

CH-20-Homogeneous-Integral-equations

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