. 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 I F Q I F dx × ( .
) ( . ) , where C is an arbitrary constant. . In the integrating factor e Pdx ò , P is the coefficient of y in the differential equation provided the coefficient of dy dx is unity.
Ordinary Differential Equations . A first order differential equation of the form dx 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 Qe Pdy Pdy Example .
Solve dy e x Given that dy + = e x ... ( ) This is a linear differential equation in y of the form dy Py = Q . Here P = ; Q e x − . Pdx