In Problems 1 -4 , find a formal solution to the vibrating string problem governed by the given initial-boundary value problem.
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Chapter 10 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
- 2. (Theorem 2.1, Eq. 2.5) By employing change of variables r = ax +y and s= bx +y, prove that the PDE: A Urr + Bury + С Uyy has the following general solution: when: = :0 u(x, y) = F(ax + y) +G(bx + y) A ‡0, B²-4AC #0arrow_forward2. Solve the following Cauchy problem: U -00 0, u(x, 0) = x – a², u(x,0) = 4x, -∞ < r < oo. Uu =arrow_forward4. Find the solution of the vibrating string problem 0²W ᎧᎳ + əx² dy² with the boundary conditions W (0, y) = W(1, y) = 0, 1 ᎧᎳ =cc 2 ду W(x, 0) = = 0, 0≤x≤1, 0≤ y ≤ 1, ㅠ cos( 2πx) - 3 sin(5x), 2 Trigonometric formula cos(A + B) = cos A cos B 3π -(x,0) = = cos(- +3πx). 2 sin Asin B.arrow_forward
- Ex. 3. Find a solution of Laplace's equation, u +u =0, inside the rectangle 0arrow_forward6 a) Consider the system of equations 1 3 Y Yit) -e dY x(t) dt 1 -1 The origin is a. Source b. Sink c. Saddle Point d. None of the abovearrow_forward1. Solve for x and y in xy + 8 + j(x²y + y) = 4x + 4 + j(xy² + x) A. 2, 2, B. 2,3 C. 3, 2 2. Determine the principal value of (3 + j4)¹ +² +j2 A. 0.42+j0.56 C. -0.42-j0.66, B. 0.42+j0.66 D. 0.42-j0.66 3. Using the properties of complex numbers. determine the two square roots of 3-j2 A. +1.82+j0.55, C. 1.82 + j0.55 B. +1.82±j0.55 D. +1.82 + j0.55 4. Evaluate: BE CALC 3-14 3+14 + 3+j4 3-j4 A. 2.44 +j4/ B. 2.44-j4 C. -2.44 + j4 D. 2.44 +j5 Evaluate log; (3 + j4). A. 0.6+j1.02 C. -0.6-j1.02 B. -0.6+j1.02 D. 0.6-j1.02, 6. The following three vectors are given; A = 20 +j20, B = 30/120° and C= 10+ j0, find AB/C C. 95/-50° B. 85-75% A. 70/45° D. 75/70" 7. If 100+5x/45° = 200/-e. Find x and 8. A. 24. 23.28 B. 23.28. 32.3° C. 23.28. 24.3% D. 23, 42.8° 8. Determine the principal value of cosh' (j0.5). A. In (1+j5) C. In j5 B. In (1± √5), D. In j(1 + √5) 2 5 1 = 9. In A-2B-C=0. if A= 2B-C-0. if A- and B-₁ find C |² -1 3 2 3 8 -3 8 3 91 C. A. 3 0 0 -3 -8 -8 -3 3 D. B. | 3 0 -3 10. Solve for a and b…arrow_forward15.17 Consider the system of equations y? + 2u? + v? – xy = 15, 2y? + u? + v + xy = 38, 1, v = -1. Think of u and v as exogenous and x at the solution x = 1, y = 4, u = and y as endogenous. Use calculus to estimate the values of x and y that correspond to u = .9 and v = -1.1.arrow_forward1.)Solve the following Partial Deferential Equations:arrow_forwardPlease provide Handwritten answerarrow_forwardThe equation of a line in the plane is ax + by + c = 0. Given two points on the plane, show how to find the values of a, b, c for the line that passes through those two points. You may find the answer to question 3 useful herearrow_forward2. A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of the time that the walk-up window is in use. Then the set of possible values for (X, Y) is the rectangle D = {(x, y): 0 ≤ x ≤ 1,0 ≤ y ≤ 1}. Suppose the joint probability density function of (X, Y) is given by: fx,y(x,y) = { / (x + y ² ) ²²(x + y²) 0≤x≤ 1,0 ≤ y ≤ 1 Otherwise 0 (a) Show that fx,y is legitimate (b) Find the probability that neither facility is busy more than one-quarter of the time. (c) Find fx and fy, the marginal probability density function of X and Y respectively. (d) Construct the conditional probability density function of Y given that X=0.8. (e) Evaluate the probability that the walk-up facility is busy at most half of the time given that X=0.8 (f) Calculate the expected proportion of the time that the walk-up facility…arrow_forward6.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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