1. Go through the Chapter 1 Powerpoint Slide (updated version, 8E-Ch01_rev2). 2. The system in Fig. 1 can be described by the following differential equation: x(t) + 3x(t) = u(t) where u(t) = sin (t) and x(0) = 1. input u(t) system 3. Solve the following ODE Fig. 1 Please work out the solution for x(t). (1) Find X₁ (t) for x(t) + 3x(t) = 0. (homogeneous solution) (2) Find xp (t) for x(t) + 3x(t) = sin(t). (nonhomogeneous solution) output x(t) Hint: xp (t) = A sin(t) + B cos(t) (why?) (3) Use the initial condition x(0)= 1 to find the general solution x(t) = xh (t) + xp (t) (1) Use control input u(t)= e-t (2) Use control input u(t) = e-³t x(t) + 3x(t) = u(t) where x(0) = 1.
1. Go through the Chapter 1 Powerpoint Slide (updated version, 8E-Ch01_rev2). 2. The system in Fig. 1 can be described by the following differential equation: x(t) + 3x(t) = u(t) where u(t) = sin (t) and x(0) = 1. input u(t) system 3. Solve the following ODE Fig. 1 Please work out the solution for x(t). (1) Find X₁ (t) for x(t) + 3x(t) = 0. (homogeneous solution) (2) Find xp (t) for x(t) + 3x(t) = sin(t). (nonhomogeneous solution) output x(t) Hint: xp (t) = A sin(t) + B cos(t) (why?) (3) Use the initial condition x(0)= 1 to find the general solution x(t) = xh (t) + xp (t) (1) Use control input u(t)= e-t (2) Use control input u(t) = e-³t x(t) + 3x(t) = u(t) where x(0) = 1.
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
Transcribed Image Text:1. Go through the Chapter 1 Powerpoint Slide (updated version, 8E-Ch01_rev2).
2. The system in Fig. 1 can be described by the following differential equation:
x(t) + 3x(t) = u(t) where u(t) = sin (t) and x(0) = 1.
input u(t)
3. Solve the following ODE
system
Fig. 1
Please work out the solution for x(t).
(1) Find X₁ (t) for x(t) + 3x(t) = 0. (homogeneous solution)
(2) Find xp (t) for x(t) + 3x(t)= sin(t). (nonhomogeneous solution)
Hint: xp (t) = A sin(t) + B cos(t) (why?)
(3) Use the initial condition x(0) = 1 to find the general solution
x(t) = xh (t) + xp (t)
(1) Use control input u(t)= e-t
(2) Use control input u(t)= e-³t
output x(t)
x(t) + 3x(t) = u(t) where x(0) = 1.
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