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We solve the partial differential equation (PDE) using the separation of variables method. ### Given PDE: \[ \frac{\partial u}{\partial x} = 2\frac{\partial u}{\partial t} + u \] with the initial condition: \[ u(x,0) = 3 e^{-3x} \] --- ### Step 1: Assume a Separable Solution We assume \( u(x,t) \) can be written as a product of two functions: \[ u(x,t) = X(x)T(t) \] Substituting into the PDE: \[ \frac{d}{dx} (X(x)T(t)) _______ _________ __________ _____ ________ ______ _____ _______ __________ __________.
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