# Solving Linear Ordinary Differential Equations

Hello. This post will be a short guide on solving first order linear ordinary differential equations (ODEs).
It is assumed that the reader knows derivatives and integrals from calculus. Knowing product rule helps here.

Topics

The Linear Differential Equation & Theory

We start with a first order linear differential equation of the form:

Dividing both sides (all terms) by gives us:

Setting as and setting as gives us the standard form as follows:

(1)

Ultimately, we want to solve for from the standard form. The procedure in doing so is not as simple as solving separable differential equations. The procedure here will require some “clever” calculus.

From (1), we multiply both sides by an integrating factor (What is will be explained later.):

(2)

Recall that the calculus product rule is .

By setting as and recognizing that , equation (2) can be expressed as:

(3)

Taking the indefinite integral on both sides with respect to the variable in (3) gives:

(4)

The solution to the standard from (1) above is:

(5)

with as the constant of integration.

What Is The Integrating Factor ?

We have solved for but what exactly is when we multiplied both sides of the standard form? Let us investigate.

Before we obtained equation (3) above, we had . This is a separable differential equation. We solve for :

Source: http://quicklatex.com/cache3/43/ql_13b732a74dc29394e28c2436d96fce43_l3.png

Since the integral of contains a constant anyways, we do not need the constant. The integration factor is . (Recall that

The Method for Solving Linear Equations

The above was the theory of solving for . What if you don’t care too much about the theory and just want the procedure?

Here is the method:

Step a) Make sure when dealing with a first order differential equation, you have the equation in standard form as follows:

Step b) Multiply both sides (all terms) by the integration factor :

Step c) With and the left side being the result of a product rule of . Rewrite the above equation as:

Step d) Integrate both sides with respect to and solve for to obtain:

Alternate Method

This alternate method involves some memorization. If you can memorize the standard form from step a), the integrating factor , and the formula in step d) then you can avoid the product rule derivation steps in b) and c).

Example

Solve for in the differential equation

Solution:

Dividing every term by and rearrangement gives us the standard form a) as follows:

We have and . Our integrating factor would be .

We use the formula in d) to obtain the solution for :

Source: http://quicklatex.com/cache3/04/ql_32fa6f74a8c1458c8b074b3595641804_l3.png

We have the general solution to . Now we find the constant in such that it satisfies . (Note that ).

Source: http://quicklatex.com/cache3/ff/ql_37c529f1357e1fb118596b3c4fa988ff_l3.png

With , we check our solution to see if it does satisfy .

Check

Notes

• The example above is a simple one to illustrate how to obtain . Other problems involve more computations.
• I have used and interchangeably.
• Don’t forget the constant of integration ! However, the constant of integration is not needed in the integration factor .
• Differential Equations is tricky as it takes a lot of work and one mistake can destroy everything.

Reference

Nagle R.K. et al., Fundamentals of Differential Equations and Boundary Value Problems – 5th Edition.

The featured spiral like image is taken from http://www.thegreatcourses.com/courses/mastering-differential-equations-the-visual-method.html.