Solution to inhomogenous linear equation

[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.

Your help is much appreciated.

thanks a lot.

 

[╔(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 ?

 

[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?

 

[Mark44]

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.

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)?

 

[whyayeman]

Mark44,

beautifully explained.

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

 

[Mark44]

Mark44,

beautifully explained.

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

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

 

[cpyap]

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