MATH 467 Partial Differential Equations Exercises - Millersville ...

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10. Find the general solution for the following first-order partial differential equation. ux −y 2 uy −yu = 0 11. Find the general solution for the following first-order partial differential equation. ux +yuy +xu = 0 12. Find the general solution for the following first-order partial differential equation. xux +yuy +2 = 0 13. For the following first-order linear partial differential equation find the general solution and the solutions satisfying the side conditions. (a) u(x,x) = x 2 (b) u(x,−x) = 1−x 2 3yux−2xuy = 0 (c) u(x,y) = 2x on the ellipse 2x 2 +3y 2 = 4 14. For the following first-order linear partial differential equation find the general solution and the solutions satisfying the side conditions. (a) u(x,−6x+2) = e x (b) u(x,−x 2 ) = 1 (c) u(x,−6x) = −4x ux −6uy = y 15. For the following first-order linear partial differential equation find the general solution and the solutions satisfying the side conditions. (a) u(x,3x) = cosx (b) u(x,2x) = x (c) u(x,x 2 ) = 1−x 4ux +8uy −u = 1 16. For the following first-order linear partial differential equation find the general solution and the solutions satisfying the side conditions. −4yux +uy −yu = 0
(a) u(x,y) = x 3 on the line x+2y = 3 (b) u(x,y) = −y on y 2 = x (c) u(1−2y 2 ,y) = 2 17. For the following first-order linear partial differential equation find the general solution and the solutions satisfying the side conditions. yux+x 2 uy = xy (a) u(x,y) = 4x on the curve y = (1/3)x 3/2 (b) u(x,y) = x 3 on curve 3y 2 = 2x 3 (c) u(x,0) = sinx 18. For the following first-order linear partial differential equation find the general solution and the solutions satisfying the side conditions. (a) u(x,4x) = x (b) u(x,y) = −2y on curve y 3 = x 3 −2 (c) u(x,−x) = y 2 y 2 ux +x 2 uy = y 2 19. Show that if u(x,y) = x 2 +y 2 tan −1 (y/x) then 20. Determine the value of n so that solves the PDE xux +yuy −u = 0. u(x,y) = x 3 tan −1 2 2 x −xy +y x 2 +xy +y 2 xux +yuy −nu = 0. 21. Let F and G be arbitrary differentiable functions and let u(x,y) = F(y/x)+xG(y/x). Show that u(x,y) solves the PDE x 2 uxx +2xyuxy +y 2 uyy = 0. 22. Let F, G, and H be arbitrary differentiable functions and let u(x,y) = F(x − y) + xG(x−y)+x 2 H(x−y). Show that u(x,y) solves the PDE uxxx +3uxxy +3uxyy +uyyy = 0.
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Page 5 and 6: 27. A square plate of edge length a
Page 7 and 8: 41. Suppose u(x,t) solves utt = a 2
Page 9 and 10: 50. Solve the boundary value proble
Page 11: on the solid cube where 0 ≤ x ≤

MATH 467 Partial Differential Equations Exercises - Millersville ...

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