Linear and Non-linear Equations (QM)

[Luna Lunaticus]

1. Problem
Recall that we defined linear equations as those whose solutions can be superposed to find more solutions. Which of the following differential/integral equations are linear equations for the function u(x,t)? Below, a and b are constants, c is the speed of light, and f(x,t) is an arbitrary function of x and t.

Homework Equations

Do not try to solve this excercise using knowledge of "differential equations etc etc etc". APPLY "linearity properties".

You check it by the same way for all the equations: substitute in u(x,t) with G(x,t)+H(x,t) and see if the end result is L(G+H)=L(G) + L(H). Then check by substituting in a*u(x,t) where “a” is a constant to see if you get L(a*u)=a*L(u).If you have L(G+H)=L(G)+L(H) and L(a*u)=a*L(u), then it must be linear.

The Attempt at a Solution

I'm very slow at calculus, so I was asking for some help to solve this while learning a bit of MathLab.

 

[Luna Lunaticus]

So, I got that 1,2,3 and even 6 are linear. 4 is non-linear because of -f(x,t) which repits itself and prevents it from be L(G+H)=L(G)+L(H). Am I right?

My last problem is with the definite integral in (5). I start to think I should paid more attention to calculus class.

 

[George Jones]

My last problem is with the definite integral in (5). I start to think I should paid more attention to calculus class.

Does (when all the integrals exist)

$$\int \left[ f \left(x\right) + g \left(x\right) \right] dx = \int f \left(x\right) dx + \int g \left(x\right) dx ?$$

 

[Luna Lunaticus]

That actually helped me a lot! Thanks. But I had to watch out with (6) though... the product of two factors of u(x,t) in the first term is explicitly nonlinear, I did not realize before.

So, the answer is: 1, 2, 3 and 5 are linear, while 4 and 6 are non-linear.

Thanks for the help!