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wish to word process your solutions, simplyate your own template so that it is clear which part of your solution refers tohich part of the assignment. Use the method of Separation of Variables to derive solutions to (2)-(5) having the form u(x, t)=\sum_{==\mathbf{i}}^{\infty} A_{n} \sin \frac{n \pi x}{L} \cos \frac{n \pi c t}{L} where A,, n= 1,2,... are constants that need to be determined in order that u(x, t)also satisfies the constraint of (6). (Note: You may assume that the constant valuereferred to by k in lecture notes or by a in the handout PDE's I: The Wave Equation'is negative.) The left-hand side is now a function of t only and the right-hand side is a function of x only. It follows that both sides must equal the same constant value k say. Let us first analyze the equation \frac{x^{8}}{x}=k \Leftrightarrow x^{\prime \prime}-k x=0 We only consider the case where k < 0. Let us first note that Equation (4) implies that X (0)T(t) = X(L)T(t) = 0. Since we are not interested in the trivial solution where T(t) = 0, we deduce that fill in boxes X (0) = X(L)=--- Since k < 0, we set k = -p,: Equation (10) has the general solutionput general solution into box below

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