If u = (x, t) is the solution of the initial value problem

satisfying |u(x. t)| <  for all x ∈ ℝ and t > 0, then

Explanation:

Given 

 with the initial condition

which is a heat equation for infinite domain.

So solution is

u(x, t) = 

Here f(x) = sin 4x + x + 1 and c = 1 then

u(x, t) = ....(i)

(1):  = 

Let  so

 = 

= ......(ii)

and  = 

Let  so

 = 

 = .....(iii)

adding (ii) and (iii) we get

 = 

                                 = 

                               =  (let u = 2p then du = 2dp)

                              = 

                             = 2

(1) is true.

From (ii) and (iii) we can see that (2), (3), (4) are false.