# Solution to inhomogenous linear equation

1. Sep 1, 2010

### whyayeman

How do I show the difference of two solutions of an inhomogenous linear equation Lu=g with the same g is a solution of the homogenous equation Lu=0.

thanks a lot.

2. Sep 1, 2010

### ╔(σ_σ)╝

Can you please clarify what you mean. What two solutions ?

Are you saying you have a system where Lu=g and Lu = 0 ?

3. Sep 1, 2010

### whyayeman

Hi thanks for the response.

Yes, there are two systems Lu= 0 and Lu=g. I read in a book that the consequence of linearity is that if you add a homogenous solution to and inhomogenous solution , you get an inhomogeneous solution, you get an inhomogenous solution. They have not explained why?

4. Sep 1, 2010

### Staff: Mentor

This is pretty straightforward. Assume that u1 and u2 are solutions to to the nonhomogeneous linear differential equation Lu = g.

What can you say about L(u1 - u2)?

5. Sep 1, 2010

### whyayeman

Mark44,

beautifully explained.

so L(u1 - u2) = L(u1) - L(u2) = 0 ? Am I right?

6. Sep 1, 2010

### Staff: Mentor

Yes. Make sure that you add what this says about u1 - u2.

7. Sep 2, 2010

### cpyap

The solution to inhomogeneous equation of
L$$\underline{u}$$=$$\underline{g}$$
is the parametric solution of L$$\underline{u}$$=$$\underline{0}$$ + $$\underline{g}$$