Advanced Math Solutions – Ordinary Differential Equations Calculator, Linear ODE

Ordinary differential equations can be a little tricky. In a previous post, we talked about a brief overview of ODEs. In this post, we will focus on a specific type of ODE, linear first order differential equations. A linear first order differential equation is an ODE that can be put in the form

\frac{dy}{dx}+p(x)y=q(x)

where p and q are continuous functions on a given interval.  In linear first order differential equations, we can derive a formula for the solution above.


How to solve a linear first order differential equation:

1. Put the equation into form \frac{dy}{dx}+p(x)y=q(x)
• This step is very important. You cannot solve, unless the equation is in this form.
2. Set p(x)=(\ln(μ(x)))' and solve for μ(x)
• μ(x) is known as the integrating factor
• Integrate both sides
• Don’t forget the unknown constant, c, and when you can combine multiple unknown constants
3. Multiply the equation in Step 1 by μ(x) and simplify
4. Use the product rule, f\cdot g^'+f^'\cdot g on the left side of the equation to make the left side of the equation equal to (μ(x)y(x))' and write it as that:
   
   μ(x)\frac{dy}{dx}+μ(x)p(x)y=μ(x)q(x)\quad\rightarrow\quad(μ(x)y(x))^'=μ(x)q(x)
5. Integrate and solve for y(x)
• Don’t forget the unknown constant and when you can combine multiple unknown 

We will go through this first example step by step (click here for a more detailed step by step): 

1. Put the equation in the form of a first order linear ODE

2. Solve for µ(x)

We integrate both sides to find µ(x).

3. Multiply by µ(x) and simplify

4. Make the left side equal to the product rule

5. Integrate and solve for y(x)

Here’s another example (click here):

Last example (click here):

Linear first order differential equations may look intimidating at first sight. These problems do take some time to solve, but are not very hard. The most important thing to remember is to not forget about the unknown constant, c.

Until next time,

Leah