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Please answer problem e

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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Please answer problem e
2 12 130% Problem 3. Consider the second-order linear ODE: x"(t) + 64x(t) = 0 where x(0) = ? and x'(0) =. (a) Using the substitution x1(t) = x(t) and x2(t) = x'(t), rewrite the second order equation as a system of linear first-order differential equations. Write your final system in matrix/vector form. (b) Solve this system by finding eigenvalues and eigenvectors of the matrix, and then reexpress your solutions so that your final solution contains only real numbers. Express the general so- lution of your system in matrix form, and the particular solution using the initial conditions above. (c) Graph the phase portait of your vector solution on the x1x2 axis. (d) Interpret your solution the initial conditions to find the solution to the second-order ODE x(t) exactly. (e) Verify that your solution is correct by showing that x1 (t) = x(t) satisfies the original second- order differential equation and its initial conditions. P Type here to search A esc 12/A 4 5 8 9 tob Q E R U cops lock A D F G H K C V alt alt ctrl
Transcribed Image Text:2 12 130% Problem 3. Consider the second-order linear ODE: x"(t) + 64x(t) = 0 where x(0) = ? and x'(0) =. (a) Using the substitution x1(t) = x(t) and x2(t) = x'(t), rewrite the second order equation as a system of linear first-order differential equations. Write your final system in matrix/vector form. (b) Solve this system by finding eigenvalues and eigenvectors of the matrix, and then reexpress your solutions so that your final solution contains only real numbers. Express the general so- lution of your system in matrix form, and the particular solution using the initial conditions above. (c) Graph the phase portait of your vector solution on the x1x2 axis. (d) Interpret your solution the initial conditions to find the solution to the second-order ODE x(t) exactly. (e) Verify that your solution is correct by showing that x1 (t) = x(t) satisfies the original second- order differential equation and its initial conditions. P Type here to search A esc 12/A 4 5 8 9 tob Q E R U cops lock A D F G H K C V alt alt ctrl
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