Non-homogeneous Differential Equations

Generally, a non-homogeneous, 2nd order, linear, ordinary differential equation can be written in the form...

Equation (1) is deemed to be non-homogeneous because the term on the right-hand side of the equation is non-zero. That is...

When , equation (1) because the homogeneous ODE...

On any given open interval, let the solution of (2), the homogeneous solution be . Let be a "particular solution" of the non-homogeneous equation (1). The general solution to (1) is given by...

So, the general solution is the superposition of 2 solutions. How?

Well, to prove this, suppose we have 2 solutions of equation (1) and we name them and . Now let the difference between the solutions be...

...and let's substitute this result into (2)...

Therefore, we can conclude that the difference of 2 solutions to the non-homogeneous equation is the solution to the homogeneous equation. That is...

And rearranging a little, we have...

Now suppose that is the general solution of (1), that is...

...and is a particular solution of (1) , we have equation (3)...

As a final check, we substitute (3) into (1)...

Now, the general solution includes all solutions. From post #1, homogeneous solution is a combination of 2 linearly independent solutions...

Substituting this into (3), and the general solution is...

...where A and B are arbitrary constants of the homogeneous solution to be determined by the initial conditions.

So finally, the process in finding solutions to 2nd order, non-homogeneous ODE's is...

1. Find the general solution of the homogeneous equation
2. Find a particular solution to the non-homogeneous equation

Get it? I know, it sounds like a dog chasing its tail in that to find a solution to the non-homogeneous equation, we need to find a solution to the non-homogeneous equation.

But we'll make sense of this and go the steps to finding such solutions in the next post.

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Credits:

All equations in this tutorial were created with QuickLatex

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First Order Differential Equations

1. Introduction to Differential Equations - Part 1
2. Differential Equations: Order and Linearity
3. First-Order Differential Equations with Separable Variables - Example 1
4. Separable Differential Equations - Example 2
5. Modelling Exponential Growth of Bacteria with dy/dx = ky
6. Modelling the Decay of Nuclear Medicine with dy/dx = -ky
7. Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky
8. Mixing Salt & Water with Separable Differential Equations
9. How Newton's Law of Cooling cools your Champagne
10. The Logistic Model for Population Growth
11. Predicting World Population Growth with the Logistic Model - Part 1
12. Predicting World Population Growth with the Logistic Model - Part 2
13. What's faster? Going up or Coming Down?

First order Non-linear Differential Equations

1. There's a hole in my bucket! Let's turn it into a cool Math problem!
2. The Calculus of Hot Chocolate Pouring!
3. Foxes hunting Bunnies: Population Modelling with the Predator-Prey Equations

Second Order Differential Equations

1. Introduction to Second Order Differential Equations
2. Finding a Basis for solutions of Second Order ODE's
3. Roots of Homogeneous Second Order ODE's and the Nature their Solutions
4. Modelling with Second Order ODE's: Undamped Free Oscillations
5. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 1
6. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 2
7. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3
8. Non-homogeneous Differential Equations

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