Question

This is a question about Ordinary Differential Equations.

For solving linear differential equations, I have seen people use the method of integrating factors and the method of variation of parameters.

Is it true that either of these 2 methods can be used to solve any linear differential equation?

If so, could you show me an example where a linear differential equation is solved using both of these methods.

If not, could you explain using examples as to why this is the case? For example, does it depend on the complicatedness of the equation etc.

Thank you.

Answer 1

For solving a linear differential equation , whether to use method of integrating factors and the method of variation of parameters , we should consider some of the conditions like -

To apply method of integrating factors , the linear differential equation should be of a first order linear differential equation .

While method of variation of parameters can be applied to both first order linear differential equation and higher order linear differential equation to find the solution of the given differential equation .

Also , if the given first order differential equation is not exact , then we use method of integrating factor to find its solution . In this method , the differential equation is multiplied with the integrating factor so as to make it exact and then find its solution .

For example -

Consider the example ,

Since this is the first order linear differential equation , therefore this can be solved using both integrating factor method and variation of parameters method .

INTEGRATING FACTOR METHOD :

Here,

Integrating factor ,

multiplying the differential equation by integrating factor we get ,

VARIATION OF PARAMETERS :

Let the solution of the given equation be in the form of -

Thus , the solution will look like -

Finally , we substitute ,

The solution is then ,

Thus , we see the solution of the differential equation is same by both the methods .

Thus , we can say that both methods of solving a differential equation can be used but the method of integrating factor has limited application as we discussed initially .