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2. Solve the initial value problem z""+x" - 4x' - 4x = 0, x(0) = 1, x'(0) = 0, x" (0) = -1. Hint: Guess one eigenvalue.

2. Solve the initial value problem z""+x" - 4x' - 4x = 0, x(0) = 1, x'(0) = 0, x" (0) = -1. Hint: Guess one eigenvalue.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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130 3 a) x"" + x' = 0. b) x'"" + x² = 1. c) x + x = 0. EXERCISES 1. Find the general solution of the following differential equations. 2. Second-Order Linear Equations 4.5 d) x"" - x' - 8x = 0. e) x'"+x" = 2et + 3t². f) x"" - 8x = 0. 2. Solve the initial value problem x"" +x" - 4x'- 4x = 0, x(0) = 1, x'(0) = 0, x" (0) = -1. Hint: Guess one eigenvalue. 3. Write a linear, fourth-order differential equation whose general solution is x(t) = C₁ + c₂t + est (c4 cos 2t + c5 sin 5t). 5t 4. What is the general solution of a fourth-order differential equation if the four eigenvalues are λ = 3 +i, 3+i? What is the differential equation? 2.6 Steady-State Heat Conduction* Let us consider the following problem in steady-state heat conduction. A cylin- drical, uniform, metallic bar of length L and cross-sectional area A is insulated on its lateral side. We assume the left face at x = 0 is maintained at To degrees L is held at TL degrees. What is the tempera- equilibrium? Here u(x) and that the right face at x = ture distribution u = u(x) in the bar after it comes to represents the temperature of the entire cross-section of the bar at position 2, where 0<x< L. We are assuming that hout along the har iel al direction
Transcribed Image Text:130 3 a) x"" + x' = 0. b) x'"" + x² = 1. c) x + x = 0. EXERCISES 1. Find the general solution of the following differential equations. 2. Second-Order Linear Equations 4.5 d) x"" - x' - 8x = 0. e) x'"+x" = 2et + 3t². f) x"" - 8x = 0. 2. Solve the initial value problem x"" +x" - 4x'- 4x = 0, x(0) = 1, x'(0) = 0, x" (0) = -1. Hint: Guess one eigenvalue. 3. Write a linear, fourth-order differential equation whose general solution is x(t) = C₁ + c₂t + est (c4 cos 2t + c5 sin 5t). 5t 4. What is the general solution of a fourth-order differential equation if the four eigenvalues are λ = 3 +i, 3+i? What is the differential equation? 2.6 Steady-State Heat Conduction* Let us consider the following problem in steady-state heat conduction. A cylin- drical, uniform, metallic bar of length L and cross-sectional area A is insulated on its lateral side. We assume the left face at x = 0 is maintained at To degrees L is held at TL degrees. What is the tempera- equilibrium? Here u(x) and that the right face at x = ture distribution u = u(x) in the bar after it comes to represents the temperature of the entire cross-section of the bar at position 2, where 0<x< L. We are assuming that hout along the har iel al direction
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