First order ODE with Dirac delta funtcion I am looking for a direct method to solve this first order ODE with Dirac delta funtcion

$$\frac{dU(t)}{dt}+k^2U(t)=\frac{1}{\sqrt{2\pi}}\delta(t)$$

with the initial condition $U(0)=\frac{1}{\sqrt{2\pi}}$.

The solution to this problem is $$U(t)=\frac{1}{\sqrt{2\pi}}e^{-k^2t}$$

My try

The integrating factor for this ode is
$$I=e^{\int k^2 dt }=e^{ k^2 t  }$$

then multiplying both sides of the differential equation by  $\,\,e^{ k^2 t  }$, we get

$$\frac{d }{d t}\left(e^{ k^2 t  } U(t)\right)=\frac{1}{\sqrt{2\pi}}e^{ k^2 t  } \delta(t)$$
Integrating both sides, we have

$$e^{ k^2 t  } U(t)=\frac{1}{\sqrt{2\pi}}\int{e^{ k^2 t  } \delta(t)}dt+C$$

From here on, I am lost. Any suggestions? 

Back ground of this problem

The above ode we got after applying Fourier transform to the following PDE
$$u_{t}=u_{xx}+\delta{(x)}\delta{(t)}$$
with $u(x,0)=\delta(x)$.
This maybe an answer to my question but needs your conformation/validation.

Continuing from where I left
$$e^{ k^2 t  } U(t)=\frac{1}{\sqrt{2\pi}}\int{e^{ k^2 t  } \delta(t)}dt+C$$
$$\Rightarrow U(t)=\bigg[\frac{1}{\sqrt{2\pi}}\int{e^{ k^2 t  } \delta(t)}dt+C\bigg]e^{ -k^2 t  }$$
Now recalling some properties of the Dirac delta function 


$$\int_{-\infty}^{\infty}f(t)\delta(t)dt=f(0)$$
$$\int_{a}^{b}f(t)\delta(t)dt=
\begin{cases}
f(0) &amp; \text{if } a&lt;t&lt;b\\
0 &amp; \text{if } x=0
\end{cases}
$$
Now since we are dealing with an IVP, where $t\geq0$, then adopting the above properties "we can have"
$$\int{e^{ k^2 t  } \delta(t)}dt=e^{0}=1,$$


which gives 
$$U(t)=\bigg[\frac{1}{\sqrt{2\pi}}+C\bigg]e^{ -k^2 t  }$$
Finally utilizing the initial condition $U(0)=\frac{1}{\sqrt{2\pi}}$, gives $C=0$, and hence
$$U(t)=\frac{1}{\sqrt{2\pi}}e^{ -k^2 t  }$$

Is this correct?
I wonder if you have a redundant initial condition? What is normally seen is the following:

U’+k^2*U=dirac(t)/sqrt(2pi)

U(0)=0

This system is equivalent to the following:

U’+k^2*U=0, t>0

U(0)=1/sqrt(2pi)

So I guess your ODE should be instead:

U’+k^2*U=dirac(t)/sqrt(2pi)

U(0)=0