10.7 First Order Linear Differential Equations

A first order differential equation of the form Py = Q . ... ( ) where P and Q are functions of x only. Here no product of y 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 solve ( ), let us consider the homogeneous equation dy Py = . ...( ) The equation ( ) can be solved as follows: Separating the variables, dy y = − Pdx . On integration, we get ye Pdx ò = C . Now, d ye Pdx   = e y Pe Pdx Pdx = e Py Qe Pdx Pdx  = ...

( ) (using ( )) Integrating both sides of ( ) with respect to x , we get the solution of the given differential equation as ye Pdx ò = Qe Pdx Here e Pdx ò is known as the integrating factor (I.F.) of ( ). Remarks . The solution of linear differential equation is