Quiz 1 Solutions - Partial Differential Equations | MTH 3326, Quizzes of Differential Equations

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Math 3326 Quiz #1 SPRING SEMESTER 2009 Name SOLUTIONS 1. Suppose that /(2) and g(y) are arbitrary differentiable functions of one variable. Verify that u(x, y) = f(a)g(y) is a solution of the second-order (nonlinear) PDE. Ullzy — Uslly = 0. At by) = Flx)ghy) Lm F U, Uy )= dH t)gly) “4 ‘ / ‘ Uy tey)=F&)gly) — ~ F's Wah Heat tog Ung bey )= Fg fy) a 2. Consider the first-order PDE u, + wy = 1. (a) Find the characteristic equation and the transformed equation associated with this PDE. From this, find the general solution to this PDE. Simplify your answer as much as possible, the characterishe equations AQ =} ss Max ee. Ket yecyd= yx 4 SixuyJ=x so T= ss ef I jptbe the tromefrmed Lquaton is -We=1 se wisy)=5+ Fy) dence He general solutions te Utvy=) ic UX) = *+ Fl4-%) (b) Now find the solution to this same PDE when we additionally require that u(0, y) = y. Again, simplify your solution as much as possible. Avon Hock Uley= X+ Fly) omd we Want y= UlO,y), Ave See Hat = ulo,y) = £f p= Fy) Hence. Uli) = x+4{ yy) = x+( ras 4 = 4+Uy = Y wlayd=y is fe solide te ee fo 7