MA Chapter 10 practice

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1 MA 33 Chapter 1 practice NAME INSTRUCTOR 1. Instructor s names: Chen. Course number: MA TEST/QUIZ NUMBER is: 1 if this sheet is yellow if this sheet is blue 3 if this sheet is white 4. Sign the scantron sheet. 5. Laplace transform table and a formula sheet on Fourier series and some PDEs are provided at the end of this booklet. 6. There are questions, each worth 5 points. Do all your work on the question sheets. Turn in both the scantron form and the question sheets when you are done. 7. Show your work on the question sheets. Although no partial credit will be given, any disputes about grades or grading will be settled by examining your written work on the question sheets. 8. NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper.

2 1. Find all T (t) such that u(x, t) = cos(x)t (t) is a solution of A. T (t) = c 1 e 4t + c e 4t. B. T (t) = c 1 e t + c e t. C. T (t) = c 1 cos(4t) + c sin(4t). D. T (t) = c 1 cos(t) + c sin(t). E. T (t) = c 1 cos(t) + c sin(t). 4u xx = u tt.. Find the equation for Y (y) such that u(x, y) = e λx Y (y) is a solution of A. Y n Y = B. yy λy = C. yy n Y = D. yy λy = E. yy + λy = u x + yu y =. 3. Determine whether the method of separation of variables can be used to replace the equation u xx xu tt = by a pair of ordinary differential equations. A. The equation can be replaced by X λxx =, T + λt =, where λ is some constant. B. The equation cannot be replaced by a pair of ordinary differential equations using this method. C. The equation can be replaced by X + λx =, T + λxt =, where λ is some constant. D. The equation can be replaced by X λx =, T + λt =, where λ is some constant. E. The equation can be replaced by X + λxx =, T + λt =, where λ is some constant. Page

3 4. Which is the solution for A. e 3πy sinh(3πx) B. e 3πy sin(3πx) C. sinh(3πy) sin(3πx) D. sinh(3πy) cos(3πx) E. none of the above. u x + u =, y x 1, y u(, y) =, u(1, y) = u(x, ) = sin(3πx), lim u(x, y) =? y 5. Find the eigenvalues and eigenfunctions of the given boundary value problem: (Assume λ > 1) u u + λu =, u() = u(π) = A. λ n = n π, u n (x) = sin(n + 1 )πx, n = 1,,... B. λ n = n π, u n (x) = sin(nπx), n = 1,,... C. λ n = n + 1, u n (x) = e x sin(nx), n = 1,,... D. λ n = n + 1, u n (x) = e x sin(nx), n = 1,,... E. λ n = n + 1, u n (x) = e x cos(nx), n = 1,, Solve the boundary value problem if possible: A. y(x) = cos x B. y(x) = sin x + c 1 cos x C. y(x) = cos x + (cos x)/3 D. y(x) = cos x + c 1 sin x + (cos x)/3 E. No solution y + 4y = cos x, y() =, y(π) =. Page 3

4 7. Find a, such that the functions 1 and x ax are orthogonal on the interval (, 3) A. 1 3 B. 1 3 C. D. 1 E Given that the Fourier series of period 4 for the function f(x) = x defined on (, ) is a + a n cos( nπx ). What is the value of a 3? A. π 3 8 B. π C. 8 9π D. π E. 3π 9. Let g(x) be the Fourier sine series of period 4 for the function f(x) defined on (, ) by { 1, x < 1, f(x) =, 1 x <. Determine the values of g(), g(.5) and g(5). Hint: Use the Fourier Convergence Theorem rather than finding coefficients in the series. A. 1,, 1 B., 1,.5 C., 1, 1 D. 1, 1,.5 E. 1,, 1 Page 4

5 1. The Fourier sine series of period 4 for the function f(x) defined on (, ) by { 1, < x < 1, f(x) =, 1 x <. is A. B. C. D. E. (1 cos( nπ )) nπ sin(nπx ) (1 sin( nπ )) nπ sin(nπx ) (1 ( 1) n ) nπ sin(nπx ) (1 ( 1) n ) nπ sin(nπx) (1 cos( nπ )) 1 nπ sin(nπx) 11. The Fourier series for the following periodic function: { 1, if 1 x <, f(x) =, f(x + ) = f(x) for all x, if x < 1, is 3 A. + (1 ( 1)n ) 1 sin nπx nπ 3 B. + (1 ( 1)n ) 1 cos nπx nπ C. 1 + (1 ( 1)n ) 1 (cos nπx + sin nπx) nπ D. 1 + (1 ( 1)n ) 1 (cos nx + sin nx) nπ E. ( 1) n+1 π sin nπx n Page 5

6 1. Let f(x) be defined as f(x) = { 1, x < 1, x < f(x + 4) = f(x). Denote by S N (x) the finite sum of Fourier series of f(x), that is S N (x) = a N + a n cos( nπ N x) + b n sin( nπ x). For fixed integer k, which of the following statement is correct? A. lim N S N (k + 1) = ( 1) k+1 B. lim N S N (k + 1) = ( 1) N+1 C. lim N S N (k + 1) = D. lim N S N (k + 1) = ( 1) k E. The limit of S N (k + 1) as N does not exist 13. Which u(x, t) does NOT satisfy the following equations A. u(x, t) = B. u(x, t) = e 9πt sin(3πx) C. u(x, t) = e πt sin(πx) u xx = u t, < x < 1, t > u(, t) =, u(1, t) = D. u(x, t) = e πt sin(πx) e 4πt sin(πx) E. u(x, t) = 5e 4πt sin( πx) 14. Let u(x, t) satisfy the heat equation Then u( 1, 1) = A. 1 B. C. 1 + e π D. 1 1 e π E. 1 + e π u xx = u t, < x < 1, t > u(, t) =, u(1, t) = u(x, ) = x + sin(πx). Page 6

7 15. Let a silver bar cm long be initially at uniform temperature 4 C. Suppose that at time t = the end x = is cooled to C while the end x = is heated to 6 C, and both thereafter maintained at those temperatures. For silver, the thermal diffusivity is known to be 1.71 cm /sec. Then the temperature u(x, t) at a point x centimeters from the left end, at time t seconds satisfies which of the following? (1) u(x, ) =, u(x, ) = 6 for < x < () u(, t) =, u(, t) = 6 for t > (3) u(x, ) = 4 for < x < (4) 1.71u xx = u t for < x <, t > (5) 1.71u xx = u tt for < x <, t > A. (1), (), (4) B. (1), (3), (4) C. (), (3), (4) D. (), (3), (5) E. (1), (3), (5) 16. Solve the temperature u(x, t) which satisfies the heat equation and the following initial and boundary conditions, u t = u xx, t >, < x < u x (, t) =, u x (, t) =, t > u(x, ) = 1, < x < A. u(x, t) = 1 1 B. sin(4x)(sin(4t) + 1 sin(8x) sin(8t) C. sin(x)(sin(4t) + sin(4x) sin(8t) D. E. (1 ( 1) n ) n π sin(nx)e nt (1 ( 1) n ) nπ sin(nπx 5 )e nπt 5 Page 7

8 17. Let v(x) be the steady-state solution of the heat conduction problem: u xx = u t, u(, t) = 5, u(3, t) =. What is v(1)? A. 1 B. C. 3 D. 4 E Which u(x, t) does NOT satisfy the following equations A. u(x, t) = 4u xx = u tt, < x < π, t > u(, t) =, u(π, t) = B. u(x, t) = 1 sin(4x)(sin(4t) + 1 sin(8x) sin(8t) C. u(x, t) = sin(x)(sin(4t) + sin(4x) cos(8t) D. u(x, t) = π sin(x)(sin(4t) + π sin(4x) cos(8t) E. u(x, t) = 1 sin(x)(sin(4t) + 1 sin(4x) sin(8t) 19. Find the solution u(x, t) of the wave equation 4u xx = u tt, < x < π, t >, satisfying the conditions u(, t) = u(π, t) = when t >, u(x, ) = when x π, u t (x, ) = sin x + 4 sin 4x when x π. 1 A. sin(4x)(sin(4t) + 1 sin(8x) sin(8t) B. sin(x)(sin(4t) + sin(4x) sin(8t) π C. sin(x)(sin(4t) + π sin(4x) sin(8t) sin(x)(sin(t) + π sin(4x) sin(4t) D. π 1 E. sin(x)(sin(4t) + 1 sin(4x) sin(8t) Page 8

9 . A string L = 5 meter long and fixed at both ends is giving an initial velocity of g(x) = m/s from equilibrium. The string has no initial displacement. Find an expression for the displacement function u(x, t) valid for all x in x 5 and all t. (Assume α = 1, where α is the constant in the wave equation.) A. B. C. D. E. (1 ( 1) n ) n π sin(nπx 5 ) sin(nπt 5 ) (1 ( 1) n ) n π sin(nx) sin(nt) (1 ( 1) n ) nπ sin(nπx 5 ) cos(nπt 5 ) (1 ( 1) n 4 ) sin(nx)(sin nt + cos(nαt)) n π (1 ( 1) n 4 ) n π sin(nx 5 ) sin(nt 5 ) 1. Which is the solution for A. e 3πy sinh(3πx) B. e 3πy sin(3πx) C. sinh(3πy) sin(3πx) D. sinh(3πy) cos(3πx) E. none of the above. u x + u =, y x 1, y u(, y) =, u(1, y) = u(x, ) = sin(3πx), lim u(x, y) =? y Page 9

10 Formula sheet Fourier series: For a L-periodic function f(x), the Fourier series for f is where for n = 1,,, a + a n cos nπx L + b n sin nπx L, a = 1 L L f(x)dx, a n = 1 L L f(x) cos nπx L dx, b n = 1 L L f(x) sin nπx L dx. Heat equation 1: The solution of the heat equation α u xx = u t, < x < L, t >, satisfying the (fixed temperature) homogeneous boundary conditions u(, t) = u(l, t) = for t > with initial temperature u(x, ) = f(x) has the general form u(x, t) = c n e n π α t/l sin nπx L, where c n = L f(x) sin nπx L dx. Heat equation : The solution of the heat equation α u xx = u t, < x < L, t >, satisfying the insulated boundary conditions u x (, t) = u x (L, t) = for t > with initial temperature u(x, ) = f(x) has the general form u(x, t) = c + c n e n π α t/l cos nπx L, where c n = L f(x) cos nπx L dx. Wave equation: The solution of the wave equation α u xx = u tt, < x < L, t >, satisfying the homogeneous boundary conditions u(, t) = u(l, t) = for t > and initial conditions u(x, ) = f(x) and u t (x, ) = g(x) for x L has the general form where c n = L u(x, t) = sin nπx L ( c n cos nπαt L + k n sin nπαt ) L f(x) sin nπx L dx and k n = nπα g(x) sin nπx L dx. Laplace equation: The solution of the Laplace equation u xx + u yy =, < x < a, y b, satisfying the boundary conditions u(x, ) = u(x, b) = for < x < a and u(, y) = and u(a, y) = f(y) for y b has the general form u(x, y) = c n sinh nπx b sin nπy b where c n = b sinh( nπa b ) b f(y) sin nπy b dy. Page 1

11 Figure 1: Laplace Transform Table f(t) =L 1 {F (s)} F (s) =L{f(t)} s. e at 1 s a 3. t n n! s n+1 4. t p (p> 1) 5. sin at 6. cos at 7. sinh at 8. cosh at 9. e at sin bt 1. e at cos bt Γ(p +1) s p+1 a s + a s s + a a s a s s a b (s a) + b s a (s a) + b 11. t n e at n! (s a) n+1 e cs 1. u c (t) s 13. u c (t)f(t c) e cs F (s) 14. e ct f(t) F (s c) 15. f(ct) 16. t f(t τ) g(τ) dτ ( ) 1 s c F c F (s) G(s) 17. δ(t c) e cs c> 18. f (n) (t) s n F (s) s n 1 f() sf (n ) () f (n 1) () 19. ( t) n f(t) F (n) (s) Page 11