The most up-to-date version of this collection of homework exercises can always be found at bob/math467/mmm.pdf.

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1 Millersville University Department of Mathematics MATH 467 Partial Differential Equations January 23, 2012 The most up-to-date version of this collection of homework exercises can always be found at bob/math467/mmm.pdf. 1. Find the general solution for the following first-order partial differential equation. 3u x +5u y xyu = 0 2. Find the general solution for the following first-order partial differential equation. u x u y +yu = 0 3. Find the general solution for the following first-order partial differential equation. u x +4u y xu = x 4. Find the general solution for the following first-order partial differential equation. 2u x +u y yu = 0 5. Find the general solution for the following first-order partial differential equation. xu x yu y +u = x 6. Find the general solution for the following first-order partial differential equation. x 2 u x 2u y xu = x 2 7. Find the general solution for the following first-order partial differential equation. u x xu y = 4 8. Find the general solution for the following first-order partial differential equation. x 2 u x +xyu y +xu = x y 9. Find the general solution for the following first-order partial differential equation. u x +u y u = y

2 10. Find the general solution for the following first-order partial differential equation. u x y 2 u y yu = Find the general solution for the following first-order partial differential equation. u x +yu y +xu = Find the general solution for the following first-order partial differential equation. xu x +yu y +2 = For the following first-order linear partial differential equation find the general solution (a) u(x,x) = x 2 (b) u(x, x) = 1 x 2 3yu x 2xu y = 0 (c) u(x,y) = 2x on the ellipse 2x 2 +3y 2 = For the following first-order linear partial differential equation find the general solution (a) u(x, 6x+2) = e x (b) u(x, x 2 ) = 1 (c) u(x, 6x) = 4x u x 6u y = y 15. For the following first-order linear partial differential equation find the general solution (a) u(x,3x) = cosx (b) u(x,2x) = x (c) u(x,x 2 ) = 1 x 4u x +8u y u = For the following first-order linear partial differential equation find the general solution 4yu x +u y yu = 0

3 (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) = For the following first-order linear partial differential equation find the general solution yu x +x 2 u y = 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 (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 u x +x 2 u y = y Show that if u(x,y) = x 2 +y 2 tan 1 (y/x) then xu x +yu y u = Determine the value of n so that ( ) x u(x,y) = x 3 tan 1 2 xy +y 2 x 2 +xy +y 2 solves the PDE xu x +yu y nu = 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 u xx +2xyu xy +y 2 u yy = 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 u xxx +3u xxy +3u xyy +u yyy = 0.

4 23. The length of a metal rod is not insulated, but instead radiation can take place into its surroundings. In this case the heat equation takes on the form: u t = κu xx c(u u 0 ) where u 0 is the constant temperature of the surroundings and c is a constant of proportionality. Show that if we make the change of variable u(x,t) u 0 = v(x,t)e αt where α is a suitably chosen constant, the equation above can be transformed into the form of the heat equation for a rod whose length is insulated. 24. The length of a metal rod is not insulated, but instead radiation can take place into its surroundings. In this case the heat equation takes on the form: u t = κu xx c(u u 0 ) where u 0 is the constant temperature of the surroundings and c is a constant of proportionality. Suppose u 0 = 0, the length of the bar is L = 1, the ends of the bar are kept at temperature 0, and the initial temperature distribution is given by f(x) for 0 x 1. Find u(x,t). 25. Consider the partial differential equation u xx +u xy +u yy = 0. (a) Let u(x,y) = f(x)g(y) and use the method of separation of variables to deduce (b) If f(x)g(y) 0 verify that f (x)g(y)+f (x)g (y)+f(x)g (y) = 0 f (x) f(x) = g (y) f (x) g(y) f(x) + g (y) g(y). (c) Show that if f (x) f(x) is not constant, then g (y) g(y) is constant, say λ. (d) Show that g(y) = Ce λy and show that g (y) g(y) = λ2. (e) Show that f (x)+λf (x)+λ 2 f(x) = 0. Solve this ODE for f(x) and show that ( ( λ ) ( 3 u(x,y) = Acos 2 x λ )) 3 +Bsin 2 x e λ(y x/2) 26. A square plate of edge length a has its planar faces insulated. Three of its edges are kept at temperature zero while the fourth is kept at constant temperature u 0. Show that the steady-state temperature distribution is given by u(x,y) = 2u 0 π (1 cos kπ) sin(kπx/a) sinh(kπy/a) ksinh(kπ)

5 27. A square plate of edge length a has its planar faces insulated. Three of its edges are kept at temperature zero while the fourth is kept at temperature f(x). Find the steady-state temperature distribution in the plate. 28. Find the Fourier Series for f(x) = x 2 on the interval [ L,L]. 29. Use the result above to obtain the sums of the following series: 1 = k Let f(x) = (x 2 1) 2 for 1 x 1. ( 1) k+1 k 2 = 1 = (2k 1) 2 1 = (2k) 2 (a) Find the Fourier Series for f(x) on [ 1,1]. (b) What is the minimum number of terms necessary to approximate f(x) by a finite series to within an error of 10 4? (c) Use the result above to find the sum of the following series. 31. Assuming that f(x) and f (x) are defined on [ L,L], show that f (x) is an even function if f(x) is an odd function and f (x) is an odd function if f(x) is an even function. 32. Find all the real eigenvalues of the following boundary value problem. 1 k 4 y +λy = 0 for 0 x 1 y(0) = y(1) y (0) = y (1) 33. Find all the real eigenvalues of the following boundary value problem. y +λy = 0 for 0 x π πy(0) = y(π) πy (0) = y (π)

6 34. For the boundary value problem below, find all the values of L for which there exists a solution. y +y = 0 for 0 x L y(0) = 0 y(l) = For the boundary value problem below, show that there are infinitely many positive eigenvalues {λ n } n=1 where lim n λ n = 1 4 (2n 1)2 π Show that if a / Z that for π < x < π. 37. For 0 < x < 2π show that y +λy = 0 for 0 x 1 y(0) = 0 y(1) = y (1) πcos(ax) 2asin(aπ) = 1 2a + cosx a cos(2x) a + cos(3x) a 2 e x = e2π 1 π 38. Use the result above to show that ( π cosh(π x) 2 sinhπ n=1 ) cos(nx) n sin(nx). n 2 +1 = n=1 cos(nx) n Use the result above to find the sum of the infinite series 1 n n=1 40. Use the result above to find the sum of the infinite series 1 (n 2 +1) 2. n=1

7 41. Suppose u(x,t) solves u tt = a 2 u xx with a 0. (a) Let α, β, x 0, and t 0 be constants, with α 0. Show that the function v(x,t) = u(αx+x 0,βt+t 0 ) satisfies v tt = β2 a 2 α 2 v xx. (b) Foranyconstantw,let ˆx = cosh(w)x+asinh(w)tandˆt = a 1 sinh(w)x+cosh(w)t. Show that x = cosh(w)ˆx asinh(w)ˆt and t = a 1 sinh(w)ˆx+cosh(w)ˆt. (c) Define û(ˆx,ˆt) = u(x,t) and show that u tt a 2 u xx = ûˆtˆt a 2 ûˆxˆx. 42. Find all the product solutions of the boundary value problem below. Assume k > 0. u tt = a 2 u xx ku t for 0 x L, t 0 u(0,t) = 0 u(l,t) = Consider the initial boundary value problem: u tt = a 2 u xx for 0 x L, t 0 u(0,t) = 0 u(l,t) = 0 u(x,0) = 3sin ( πx ) L u t (x,0) = 1 2 sin ( 2πx L ( ) 4πx sin L ). Find the Fourier Series solution and the solution according to D Alembert s formula and show that they are equal. 44. Solve the initial boundary value problem: u tt = a 2 u xx for 0 x π, t 0 u x (0,t) = 0 u x (π,t) = 0 u(x,0) = cos 2 x u t (x,0) = sin 2 x. 45. A string is stretched tightly between x = 0 and x = L. At t = 0 it is struck at the position x = b where 0 < b < L in such a way that the initial velocity u t is given by { v0 for x b < ǫ u t (x,0) = 2ǫ 0 for x b ǫ.

8 Find the solution to the wave equation for this initial condition. Discuss the case where ǫ Define new coordinates in the xy-plane by ˆx = ax+by +f ŷ = cx+dy +g where a, b, c, d, f, and g are constants with ad bc 0. Define û(ˆx,ŷ) = u(x,y). (a) Show that if u is C 2, then u xx +u yy = (a 2 +b 2 )ûˆxˆx +2(ac+bd)ûˆxŷ +(c 2 +d 2 )ûŷŷ. (b) Suppose that (ˆx, ŷ) are the new coordinates obtained by rotating the original axes by some angle θ in the counterclockwise direction. Verify that a = cosθ, b = sinθ, c = sinθ, and d = cosθ. Show that in this case 47. Solve the boundary value problem u xx +u yy = ûˆxˆx +ûŷŷ. u xx +u yy = 0 for 0 < x < π and 0 < y < π u(x,0) = sinx u(x,π) = sinx u(0,y) = siny u(π,y) = siny. 48. Find a function of the form U(x,y) = a + bx + cy + dxy such that U(0,0) = 0, U(1,0) = 1, U(0,1) = 1, and U(1,1) = 2. Use this function to solve the following boundary value problem. u xx +u yy = 0 for 0 < x < 1 and 0 < y < 1 u(x,0) = 3sin(πx)+x u(x,1) = 3x 1 u(0,y) = sin(2πy) y u(1,y) = y Solve the boundary value problem u xx +u yy = 0 for 0 < x < π and 0 < y < π u(x,0) = 0 u(x,π) = x(π x) u(0,y) = 0 u(π,y) = 0.

9 50. Solve the boundary value problem u xx +u yy = 0 for 0 < x < π and 0 < y < π u y (x,0) = cosx 2cos 2 x+1 u y (x,π) = 0 u x (0,y) = 0 u x (π,y) = Find the steady-state temperature distribution for an annulus of inner radius 1 and outer radius 2 subject to the boundary conditions: 52. Solve the boundary value problem 53. Solve the boundary value problem u(1,θ) = 3+4cos(2θ) u(2,θ) = 5sinθ. u xx +u yy = 0 for x 2 +y 2 < 1 u(1,θ) = 1+8cos 2 θ u(r,θ+2π) = u(r,θ). u xx +u yy = 0 for 1 < x 2 +y 2 < 2 u(1,θ) = a u(2,θ) = b u(r,θ+2π) = u(r,θ). 54. A flat heating plate is in the shape of a disk of radius 5. The plate is insulated on the two flat faces. The boundary of the plate is given a temperature distribution of f(θ) = 10θ 2 where the central angle θ ranges from π to π. What is the steady-state temperature at the center of the plate? 55. Let z = a+ib be a complex number (a and b are real numbers and i = 1). Show that sinz = sin(a)cosh(b)+icos(a)sinh(b). 56. Solve the initial boundary value problem: u t = 2(u xx +u yy ) for 0 x 3 and 0 y 5 u(x,0,t) = 0 u(x,5,t) = 0 u(0,y,t) = 0 u(3,y,t) = 0 ( ) 3πy u(x,y,0) = cosπ(x+y) cosπ(x y)+sin(2πx)sin. 5

10 57. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature 100 C. At time t = 0 the cube is placed in anenvironment whose constant temperature is 0 C. Find the temperature at the center of the cube as a function of time. 58. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature 100 C. Five faces of the cube are insulated. At time t = 0 the cube is placed in an environment whose constant temperature is 0 C. Find the temperature at the center of the cube as a function of time. 59. Solve the boundary value problem: u xx +u yy +u zz = 0 for 0 < x < π, 0 < y < π, 0 < z < π u x (0,y,z) = 0 u x (π,y,z) = 0 u y (x,0,z) = 0 u y (x,π,z) = 0 u z (x,y,0) = 0 u z (x,y,π) = 1+4sin 2 xcos 2 y. 60. Let f(x,t), g(y,t), and h(z,t) solve the respective heat equations f t = kf xx g t = kg yy h t = kh zz. Show that u(x,y,z,t) = f(x,t)g(y,t)h(z,t) solves the partial differential equation 61. Consider the partial differential equation u t = k(u xx +u yy +u zz ). u xx +u yy +u zz = 0 on the rectangular solid where 0 x L, 0 y M, and 0 z N. Suppose the values of u have been specified at the eight corners of the solid. Find a solution of the form u(x,y,z) = axyz +bxy +cyz +dxz +ex+fy +gz +h to the PDE. 62. Consider the partial differential equation u xx +u yy +u zz = 0

11 on the solid cube where 0 x 1, 0 y 1, and 0 z 1. Suppose u obeys the following boundary conditions. Find a solution of the form to the boundary value problem. u x (0,y,z) = a 0 u x (1,y,z) = a 1 u y (x,0,z) = b 0 u y (x,1,z) = b 1 u z (x,y,0) = c 0 u z (x,y,1) = c 1 u(x,y,z) = Ax 2 +By 2 +Cz 2 +Dx+Ey +Fz 63. Convert the function u(x,y,z) = 1/ x 2 +y 2 +z 2 to spherical coordinates and show that u = Convert the function u(x,y,z) = xyz to spherical coordinates and show that u = Solve the heat equation on the solid sphere of radius 1 with boundary condition u(1,t) = 0 and initial condition u(ρ,0) = sin3 (πρ). ρ