The task: solve this problem on $R\times R_{+}: $ $U_{t}=U_{xx}+cos x$, $U(0, x)=exp(2x)$
We've only solved problems like this with Poisson's formula, which requires the function $U(0,x)$ to be bounded. How this is not the case in this problem, I don't know how to proceed. The only other method we did for such equations is separating variables, but I need boundary conditions for that.
The function $v(t,x)=\exp(2x+4t)$ is clearly a solution of the homogeneous problem, so put $U=v+w$. Then $w$ solves the Cauchy problem \begin{align} w_t&=w_{xx}+\cos x,\\ w|_{t=0}&=0. \end{align} To solve this, use the method of undetermined coefficients. Seek a solution of the form $$ w(t,x)=A(t)\cos x, $$ where, to satisfy the initial condition, we want $A(0)=0$. Substituting this ansatz into the PDE for $w$ and equating the coefficient of $\cos x$ equal to zero, we obtain the Cauchy problem \begin{align} A'(t)&=-A(t)+1,\\ A|_{t=0}&=0. \end{align} Hence, $A(t)=1-e^{-t}$, and $w(t,x)=(1-e^{-t})\cos x$. This yields $U$.