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# First Order Linear Differential Equations

Mathematics: Ordinary Differential Equations: First Order Linear Differential Equations

First Order Linear Differential Equations

A first order differential equation of the form where P and Q are functions of x only. Here no product of dy/dx and its derivative dy/dx occur and the dependent variable y and its derivative with respect to independent variable x occurs only in the first degree.

To integrate (1), let us consider the homogeneous equation dy/dx + Py = 0 . ….(2)

The equation (2) can be integrated as follows: Integrating both sides of (3) with respect to x, we get the solution of the given differential equation as Here ePdx is known as the integrating factor (I.F.) of (1).

### Remarks

1. The solution of linear differential equation is

y × (I.F ) = Q(I .F )dx + C , where C is an arbitrary constant.

2. In the integrating factor e Pdx , P is the coefficient of y in the differential equation provided the coefficient of dy/dx is unity.

3. A first order differential equation of the form dx/dy + Px = Q , where P and Q are functions of y only. Here no product of x and its derivative dx/dy occur and the dependent variable x and its derivative with respect to independent variable y occurs only in the first degree.

In this case, the solution is given by xe Pdy = Qe Pdy dy + C .     Tags : Mathematics , 12th Maths : UNIT 10 : Ordinary Differential Equations
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12th Maths : UNIT 10 : Ordinary Differential Equations : First Order Linear Differential Equations | Mathematics