How to find Integrating Factors for Linear First-Order Differential Equations
Figure 1. Solutions of Equation (10)
Let's briefly recap the exponential decay model, which is typically modeled by...
Being a separable first order differential equation, we can separate the variables and get all of the y's on the left-hand side and all of the x's on the right-hand side and solve accordingly...
The general solution is...
Ok, so we should be plenty familiar with the above solution. If not, please review my tutorials on separable DE's.
Now, let's rewrite equation (1), taking all of the terms to the left-hand side...
Equation (3) is called a homogeneous, linear, first-order differential equation. It can be more generally written as...
...where P(x) can be any continuous function over a specified domain. Let's add Q(x) (another continuous function) to the right-hand side and we have...
Equation (5) is a non-homogeneous first-order differential equation, by virtue of the addition function Q(x).
For example, let's try to solve...
Unfortunately we can't use separation of variables to solve (7) because we're not able to separate the variables! We are simply not able to rearrange dy/dx into the form F(x)G(x). The equation is still soluble, but we'll need a different approach.
Ok. What approach then?
What if we multiply equation (7) by x2? We get...
What can you notice about the left-hand side of the above equation? Is it not the product rule expansion of
So equating this to the x3 on the right-hand side...
Now integrating both sides...
How cool is that? We have found a solution to a non-separable DE. And it turns out that we can solve all linear, first-order equations in the same manner by multiplying everything by a function I(x) called an integrating factor...
The question is: how do we find this integrating factor?
Well, let's refer back to equation (5). We want to multiply equation (5) by I(x), that is...
...such that the left-hand side of equation (8) becomes...
What this means is that the product IP is the derivative of I(x). So equating the coefficients of y-term...
It turns out equation (9) is a separable differential equation which we can solve...
And thus we have found a general expression to find the integrating factor. There actually are an infinite number of integrating factors, but we only need one. Which is why for the above working, we do not need to include a constant of integration C, which would have given us the most general solution for the integrating factor. We just need a basic particular solution, so we'll consider C = 0.
Therefore, to solve any linear, first-order non homogeneous differential equation of the form in equation (5), all we need to do is multiply both sides of the equation by the integrating factor
Let's do a couple of examples...
Example 1: General Solution Problem
Let's jump straight into a reasonably tricky example...
The first step is to rearrange equation (10) to the same form as equation (5)...
Now we can find the integrating factor I(x) by the formula above...
Then multiply (11) through by integrating factor and carrying through the subsequent operations...
Great. Finally, we just multiplying everything by x2 to isolate y(x) on the left-hand side...
Equation (12) is the general solution to equation (10)
A plot of the general solution for varying values of C is shown in Figure 1 above.
Example 2: Initial Value Problem
Let's solve...
With initial condition
The integrating factor for equation (13) is...
Now multiplying (13) with integrating factor and carrying through with the integration...
And applying the initial condition to find the particular solution...
And thus the particular solution is...
A plot of this very cool looking solution is shown in Figure 2 below.
Credits:
Examples 1 and 2 are my worked solutions of exercises from Advanced Engineering Mathematics 10th Edition,
Kreyszig.
All equations in this tutorial were created with QuickLatex
All graphs were created with www.desmos.com/calculator
First & Second Order Linear ODE's and their applications
- How to find Integrating Factors for Linear First-Order Differential Equations
- An Application of First Order ODEs: The RL Circuit
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