这个作业是完成拉普拉斯变换相关的数学问题
Math 123 Practice Final Exam May 12, 2020
(Please Print)
DIRECTIONS:
• Do not open Exam until instructed to do so.
• Do each of the problems and show all work.
• NO WORK MEANS NO POINTS!
• Box or circle and LABEL your final solution.
• You will have 135 minutes to complete this Exam.
SCORES:
1. /20
2. /10
3. /15
4. /15
5. /20
6. /20
7. /10
8. /10
9. /10
Total: /130
1. 20 points Solve
(a) ˙x = −x2t
(b) 2xy0 − y = x for x > 0.
2. 10 points Use the properties of matrices to prove that multiplication by a m × n matrix A
is a linear transformation from Rn to Rm.
To be more formal, let v ∈ R
n and LA(v) = Av. Explain why Av ∈ R
m and show LA(v)
satisfies the definition of a linear transformation.
3. 15 points Let A =1 2 34 5 6and B =1 23 4.
(a) Find AB.
(b) Find BA.
(c) Find B−1.
4. 15 points Let
A =0 0 0 −9 1
−2 0 0 0 0
0 0 −3 0 0
0 −1 0 0 0
1 0 0 0 6.
(a) Using any method, calculate the determinant of A, det(A).
(b) Is A invertible? Why or why not?
(c) How many solutions does Ax = 0 have? Why?
(d) Extra Credit: Calculate the adjoint (adjugate) of A.
(e) Extra Credit: Find the inverse of A.
5. 20 points Letx¨ − 2 ˙x + x = 0.
(a) Verify x1(t) = et and x2(t) = tet are solutions.
(b) Prove S = {x1, x2} is a linearly independent set of solutions.
(c) What is the dimension of the solution space?
6. 20 points LetA =−11 8−8 9.
Find the solution to˙xy= Axywith x(0) = 3 and y(0) = 4.
7. 10 points Letx¨ − x = etwith x(0) = ˙x(0) = 0. Find the impulse response and use a convolution to solve for x(t).
8. 10 points
˙xy=5 −14 1 xy, P =1 12 1. Let uw= P−1xy=−1 12 −1 xy=−x + y2x − y.
(a) Prove
˙uw= P−1˙xy=3u + w3w=3 10 3 uw.
(b) Solve the system in (a) for uw.
(c) Use the answer in (b) to find xy.
9. 10 points Two planes are given by
4x + 2y + 3z = 3 and − 3x + 4y + −3z = 4
Find the line, P0 + αv with P0, v ∈ R
3 and α ∈ R, where the two planes intersect.
1 Laplace Transforms
f(t) L{f(t)}(s) f(t) L{f(t)}(s) 11se−at 1s + at1s2tn n!sn+1 cos ω0t ss2 + ω20sin ω0t ω0s2 + ω20cosh ω0t ss2 − ω20sinh ω0tω0s2 − ω20u(t − a)e−assδ(t − a) e −as
Table 1: Laplace Transforms
Theorem 1. Let y0 + p(x)y = q(x) with p(x) and q(x) continuous. Then y(x) = Ce− R x x0 p(t) dt + e −R x x0 p(t) dt Z xx0 eR t x0 p(s) ds q(t) dt for any real number C.
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