Page 208 - Numerical Analysis Using MATLAB and Excel

Exercises

5.13 Exercises

Solve the following ODEs by any method and verify your answers with MATLAB.

2
d y dy
1. -------- + 4------ + 3y = t – 1
dt 2 dt

2
d y dy t –
2. -------- + 4------ + 3y = 4e
dt 2 dt
2
dy
d y
-- cos(
3. -------- + 2------ + y = cos 2 t Hint: Use cos 2 t = 1 - 2t + 1 )
dt 2 dt 2
2
d y
4. -------- + y = sec t
dt 2
5 Express the integro−differential equation below as a matrix of state equations where
k k and k, 1 2 , 3 are constants.

dv 2 dv t
d
-------- + k ------ + k v + k 1 ∫ vt = sin 3t + cos 3t
3
2
dt 2 dt 0
6. Express the matrix of the state equations below as a single differential equation, and let
xy() = yt() .
x · 1 0 1 0 0 x 1 0
x · 2 = 0 0 1 0 ⋅ x 2 + 0 ut()
x · 3 0 0 0 1 x 3 0

x · 4 – 1 – 2 – 3 – 4 x 4 1
7. Compute the eigenvalues of the matrices , , and below.
AB
C
0 1 0
A = 1 2 B = a0 C = 0 0 1
3 – 1 a – b
– 6 – 11 – 6

Hint: One of the eigenvalues of matrix C is 1– .

Numerical Analysis Using MATLAB® and Excel®, Third Edition 5−47

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