Word Count : 633

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We are given the following problem to solve using separation of variables: \[ u_{xt} = e^{-t} \cos(x) \] with the boundary conditions: \[ u(x, 0) = 0 \quad \text{and} \quad \frac{\partial u}{\partial t}(0, t) = 0. \] ### Step 1: Express the equation in a form that can be separated. Start by writing the given equation: \[ u_{xt} = \frac{\partial^2 u}{\partial x \partial t} = e^{-t} \cos(x). \] To solve this using the method of separation of variables, assume that the solution can be written as a product of functions, i.e., \[ u(x, t) = X(x) T(t), \] where \( X(x) \) is a function of \( x \), and \( T(t) \) is a function of \( t \). ### Step 2: Substitute into the equation. We substitute \( u(x,t) = X(x) T(t) \) into the given equation. The partial derivatives become: \[ u_{xt} = \frac{\partial}{\partial t} \left( X(x) \frac{\partial T(t)}{\partial t} \right) = X(x) T'(t), \] _________ __________ ____ _________ ______ ____ ___ _____ _______ ____.
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