MATH300 Midterm: MATH 300 UofA Exam Solution 4
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Derive the general solution of the equation a. = u, a, b 6= 0 by using an appropriate change of variables. = cx + dt, where a, b, c, and d are to be determined so as to reduce the partial di erential equation to and ordinary di erential equation, which we can then solve. U and the original partial di erential equation becomes (ab + ba) Now let b = b, a = a, c = 0, and d = 1/a, then the equation becomes. U = 0, and multiplying this equation by e , we have e u. (cid:0)e u(cid:1) = 0, and the quantity e u is independent of . Therefore, the solution is u = f ( )e , where f is an arbitrary function of . In terms of the original variables, the solution is u(x, t) = f (ax bt)et/a. Use d"alembert"s method and the superposition principle to solve the wave equation.
