Austin Cholley Jones MAT 275 ONLINE B Spring 2021
Assignment Section 7.4 Basic Theory of First order Linear systems due 04/20/2021 at 11:59pm MST
1. (1 point) Suppose
y0
1=t6y1+5y2+sec(t),
y0
2=sin(t)y1+ty2−3.
This system of linear differential equations can be put in the
form y0=P(t)y+g(t). Determine P(t)and g(t).
P(t) =
g(t) =
Answer(s) submitted:
•tˆ(6)
•sec(t)
(correct)
Correct Answers:
•[ tˆ6, 5 ] <br /> [ sin(t), t ]
•[ sec(t) ] <br /> [ -3 ]
2. (1 point) Suppose
(t+2)y0
1=6ty1+9y2,y1(1) = 0,
(t−3)y0
2=5y1+8ty2,y2(1) = 2.
(1) This system of linear differential equations can be put
in the form ~y0=P(t)~y+~g(t). Determine P(t)and ~g(t).
P(t) =
~g(t) =
(2) Is the system homogeneous or nonhomogeneous?
• Choose
• homogeneous
• nonhomogeneous
(3) Find the largest interval a<t<bsuch that a unique so-
lution of the initial value problem is guaranteed to exist.
Interval: help (inequalities)
Answer(s) submitted:
•6t
•0
•homogeneous
•(-2,3)
(score 0.5)
Correct Answers:
•<table border=’0’ cellspacing=’10’><tr><td> 6*t/(t+2) </td><td> 9/(t+2) </td></tr> <tr><td> 5/(t-3) </td><td> 8*t/(t-3) </td></tr></table>
•<table border=’0’ cellspacing=’5’><tr><td> 0 </td></tr><tr><td> 0 </td></tr></table>
•homogeneous
•-2 < t < 3
3. (1 point) Two solutions of a 2 ×2 first order linear system
are given by
y1=e5t
e5t,y2=−e−t
2e−t
Set up the Wronskian of the two solutions
Wronskian = det =
Are the solutions linearly independent or linearly dependent?
• Choose
• linearly independent
• linearly dependent
Solution:
W(y1,y2) = dety1y2=dete5t−e−t
e5t2e−t=3e4t
Since the Wronskian is non zero, the solutions are linearly inde-
pendent.
Answer(s) submitted:
•eˆ(5t)
•-eˆ(-t)
•eˆ(5t)
•2eˆ(-t)
•3eˆ(4t)
•linearly independent
(correct)
Correct Answers:
•eˆ(5*t)
•-eˆ(-t)
•eˆ(5*t)
•2*eˆ(-t)
•3*eˆ(4*t)
•linearly independent
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