A bounded solution to the partial differential equation $ \ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+e^{-t} \ $ $ \ \text{is/are}$
$(i)$ $u(x,t)=-e^{-t}$,
$(ii)$ $ u(x,t)=e^{-x} e^{-t}$,
$(iii)$ $u(x,t)=e^{-x}+e^{-t}$,
$(iv)$ $u(x,t)=x-e^{-t}$.
Answer:
This is a non-homogeneous heat equation.
We know that $ t>0$.
The general solution of heat equation is
$u(x,t)=e^{-Ct} \sin (C^2 x) $ , where $ C$ is a constant.
Thus option $(i)$ is true because it satisfies the equation as well as it has the form of general solution.
But I can not comment on other options.
Please help me.
I am assuming that $u:\mathbb{R}\times [0,\infty)\to \mathbb{R}.$ Then observe that $$u(x,t)= -e^{-t}\text{ satisfies the heat equation and }u<0\text{ and so it is bounded.}$$ $$u(x,t)= -e^{-x}e^{-t}\text{ does not satisfy the heat equation.}$$ $$u(x,t)= e^{-x} + e^{-t}\text{ does not satisfy the heat equation.}$$ $$u(x,t)= x-e^{-t}\text{ satisfies the heat equation but so it is not bounded.}$$
The key idea is to plug in the form of the function and see if it satisfies the PDE and is bounded.