Verifying if this PDE is a solution

[jc2009]

PROBLEM: Verify that the functions [x+1]e^(-t) ; e^(-2)sint ; and xt are respectively solutions of the nonhomogeneous equations
Hu = -e^(-t)[x+1] ; Hu = e^(-2x)[4sint+cost] ; and Hu = x
where H is the 1D heat operator H = $$\frac{\partial}{\partial t}$$ - $$\frac{\partial^2}{\partial x^2}$$

i did this the verification part,, the problem is with the second part of the problem
Find a solution of the PDE
Hu = $$\sqrt{2} x$$ + [Pi]e^(-2x) [4sint + cost] + e^(-t)[x+1]

isn't the first part a solution for this PDE? i don't understand the question

any hints how to setup this PDE?

 

[matematikawan]

You have already verified some particular solutions for $$Hu =\phi(x,t).$$
Note that the DE is linear nonhomogeneous.

If u₁(x,t) (resp. u₂(x,t)) is a solution of $$Hu =\phi_1(x,t)$$ (resp. $$Hu =\phi_2(x,t)$$ )
then
u₁(x,t) + u₂(x,t) will be a solution of

$$Hu =\phi_1(x,t) + \phi_2(x,t).$$