Advanced Calculus fi..

Chapter 10 Partial Differential Equations 695Show that replacement of a, by p,(t), p, by q,(t), and substitution in (10.121) lead tothe equations:sinnat + q; cos nut = 0,nap; cos nut - naq; sin nut = - Fl sin nx dx,nP 0and hence one obtains (10.122).6. Let u 1 (x, t), u2(x, t), u3(x, t), respectively, be solutions of the problems (for 0 < x < n,t > 0):4 .S"U, - U IX = F(x, t), u(0, t) = 0, u(n, t) = 0;Show that ul(x, t) + u2(x, t) + u3(x, t) is a solution of the problemu, - u,, = F(x, t), u(0, t) = a(t), ' u(n, t) = b(t).This shows that the effects of the different ways of forcing the system combine by superposition.10.13 EQUATIONS WITH VARIABLE COEFFICIENTS 8STURM-LIOUVILLE PROBLEMSIn order to determine the normal modes in the problemwe make the substitution:and are led to the equations:u = A(x) sin(ht + E )When p(x) is constant, we know that the only solutions of (10.124) are the functionsA,,(x) = c sin (nnxlL); the associated frequencies A, are of form an. What is thenature of the solutions when p is variable?A similar question arises if we consider the problem for the heat equation withvariable coefficient H(x):