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To solve these problems numerically, we first verify analytically whether the given functions satisfy the respective differential equations and then discuss numerical methods. --- ### (a) Verification for the One-Dimensional Wave Equation The one-dimensional wave equation is: \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \] For the given function: \[ u(x,t) = A(x + ct)^3 \] #### Step 1: Compute Partial Derivatives 1. First derivative with respect to \( x \): \[ \frac{\partial u}{\partial x} = A \cdot 3(x + ct)^2 \] 2. Second derivative with respect to \( x \): \[ \frac{\partial^2 u}{\partial x^2} = A ______ _______ _____ _______ ______ _______ _______ ____ __________ ________ ____ ___.
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