# Check PDE for linearity

frenchkiki

## Homework Statement

Check du/dt + d^2u/dx^2 + 1 = 0

## Homework Equations

L is a linear operator if:

cL(u)=L(cu) and L(u+v)=L(u)+L(v)

## The Attempt at a Solution

L = d/dt + d^2/dx^2 + 1

L(cu) = d(cu)/dt + d^2(cu)/dx^2 + 1 = c du/dt + c d^2(u)/dx^2 + 1 ≠ cL(u) = c du/dt + c d^2/dx^2 + c. So I found that it is not linear since it does not satisfy cL(u)=L(cu). However the solution tells me that it is. Can anyone spot my error?

Thanks

Homework Helper
Are you sure the problem wants you to include the '+1' in the definition of the operator?

frenchkiki
Are you sure the problem wants you to include the '+1' in the definition of the operator?

So that would be du/dt - d^2u/dx^2 = -1?

Then I have Lu = -1, L(cu)= d(cu)/dt - d^2(cu)/dx^2 = c du/dt + c d^2u/dx^2 = c L(u), L(u+v) = d(u+v)/dt - d^2(u+v)/dx^2 = du/dt + d^2u/dx^2 + dv/dt + d^2v/dx^2 = Lu + Lv, so it's linear and inhomogenous.

Thank you.