Page 307 - Applied Numerical Methods Using MATLAB

296 ORDINARY DIFFERENTIAL EQUATIONS
−1
X(s) = [sI − A] {x(0) + BU(s)}
1
1 s + 3 1
1
1
= +
s(s + 3) + 2 −2 s 0 0 s
1 s + 3 + 1/s
=
(s + 1)(s + 2) −2 + 1
2
(s + 3s + 1)/s(s + 1)(s + 2)
=
−1/(s + 1)(s + 2)
1/2 1 1/2 1 −t 1 −2t
X 1 (s) = + − , x 1 (t) = + e − e (P6.1.2a)
s s + 1 s + 2 2 2
−1 1 −t −2t
X 2 (s) = + , x 2 (t) =−e + e (P6.1.2b)
s + 1 s + 2
(b) Find the numerical solution of the above state equation by using the
routine “ode_RK4()” (with the number of segments N = 50) and the
MATLAB built-in routine “ode45()”. Compare their execution time
(by using tic and toc) and closeness to the analytical solution.
6.2 A Second-Order Linear Time-Invariant Differential Equation

Consider the following second-order differential equation

x (t) + 3x (t) + 2x(t) = 1 with x(0) = 1,x (0) = 0 (P6.2.1)
(a) Check the procedure and the result of obtaining the analytical solution
by using the Laplace transform technique.
1
2
s X(s) − x (0) − sx(0) + 3(sX(s) − x(0)) + 2X(s) =
s
2
s + 3s + 1 1 1
X(s) = , x(t) = + e −t − e −2t (P6.2.2)
s(s + 1)(s + 2) 2 2
(b) Define the differential equation (P6.2.1) in an M-file so that it can be
passed to the MATLAB routines like “ode_RK4()”or“ode45()”as
their input argument (see Section 6.5.1).

6.3 Ordinary Differential Equation and State Equation
(a) Van der Pol Equation
Consider a nonlinear differential equation
d 2 2 d
y(t) − µ(1 − y (t)) y(t) + y(t) = 0 with µ = 2 (P6.3.1)
dt 2 dt

Compose a program to solve this equation with the initial condition

[y(0)y (0)] = [0.50]and [−1 2] for the time interval [0, 20] and plot

y (t) versus y(t) as well as y(t) and y (t) along the t-axis.