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$\begin{aligned} & {{u}_{t}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\

& u(x,0)={{u}_{0}}, \\

& {{u}_{x}}(0,t)=u(0,t).

\end{aligned}

$

I need to apply the Laplace transform to solve it. I'll denote $u(x,s)=\mathcal L(u(x,\cdot))(s),$ so for the first line I have $s\cdot u(x,s)-u(x,0)=\dfrac{\partial^2 u(x,s)}{\partial x^2},$ now here's my problem, when I did this problem my professor told me I can't apply the transform to the condition $u(x,0)$ why? Well after this for the third line I have $\dfrac{\partial u(x,s)}{\partial x}-u(0,s)=0$ (1). So we have to solve $\dfrac{{{\partial }^{2}}u(x,s)}{\partial {{x}^{2}}}-s\cdot u(x,s)=-{{u}_{0}}$ which gives a a solution $u(x,s)=c_1e^{-\sqrt sx}+c_2e^{\sqrt sx}+\dfrac{u_0}s$ (2).

Now do I need to use (2) and (1) to find the constants? And after that I need to find the inverse Laplace transform, so far, is it correct?

Thanks!