The general solution of the differential equation e the function x(t) = C₁ cos 2t + C2 sin 2t + c3. "". +4x' = 0 happens to a) Verify that x(t) is actually a solution of this differential equation. b) Write the system of equations that determines the constants C₁, C2, nd C3 of the solution that satisfies the initial conditions x(0) = 3, x'(0) = -, x" (0) = -1. Do not solve the system. am beans. c) Write the system of equations for the constants C₁, C2, and c3 of all olutions that satisfies the boundary conditions x(0) = 5, x(π/6) = 2. Do not solve the system.
The general solution of the differential equation e the function x(t) = C₁ cos 2t + C2 sin 2t + c3. "". +4x' = 0 happens to a) Verify that x(t) is actually a solution of this differential equation. b) Write the system of equations that determines the constants C₁, C2, nd C3 of the solution that satisfies the initial conditions x(0) = 3, x'(0) = -, x" (0) = -1. Do not solve the system. am beans. c) Write the system of equations for the constants C₁, C2, and c3 of all olutions that satisfies the boundary conditions x(0) = 5, x(π/6) = 2. Do not solve the system.
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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Question
![The problem involves finding the general solution to the differential equation \( x''' + 4x' = 0 \). The given solution is:
\[ x(t) = c_1 \cos 2t + c_2 \sin 2t + c_3. \]
Tasks:
(a) Verify that \( x(t) \) is indeed a solution of this differential equation.
(b) Formulate the system of equations to find the constants \( c_1, c_2, \) and \( c_3 \) for the solution that satisfies the initial conditions \( x(0) = 3 \) and \( x'(0) = 5 \), and \( x''(0) = -1 \). Do not solve the system.
(c) Write the system of equations for the constants \( c_1, c_2, \) and \( c_3 \) for the solutions that satisfy the boundary conditions \( x(0) = 5 \) and \( x(\pi/6) = 2 \). Do not solve the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F801d34df-dcab-45aa-85e0-2349cce424a4%2Fed580437-7aa9-4819-b282-19279397aee2%2Fdkgw8q7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem involves finding the general solution to the differential equation \( x''' + 4x' = 0 \). The given solution is:
\[ x(t) = c_1 \cos 2t + c_2 \sin 2t + c_3. \]
Tasks:
(a) Verify that \( x(t) \) is indeed a solution of this differential equation.
(b) Formulate the system of equations to find the constants \( c_1, c_2, \) and \( c_3 \) for the solution that satisfies the initial conditions \( x(0) = 3 \) and \( x'(0) = 5 \), and \( x''(0) = -1 \). Do not solve the system.
(c) Write the system of equations for the constants \( c_1, c_2, \) and \( c_3 \) for the solutions that satisfy the boundary conditions \( x(0) = 5 \) and \( x(\pi/6) = 2 \). Do not solve the system.
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