Consider dM dt t 1+t subject to the condition M(0) = 0. Use the 7-step method to solve this problem and omit 'Step 7: Sketch. Clearly label each step and reduce your solution to M(t) = (1 + t)e [2 - e-'(t +2)] Notes: (i) Clearly show in 'Step 2: Integrating Factor' how P(t) = exp(t)/(1+ t), (ii) Carry out the full calculations in 'Step 4: Verify', and (iii) show all of your work when integrating-by-parts to receive points in 'Step 5: Integrate'. -M = (1 + t) ²e-²¹
Consider dM dt t 1+t subject to the condition M(0) = 0. Use the 7-step method to solve this problem and omit 'Step 7: Sketch. Clearly label each step and reduce your solution to M(t) = (1 + t)e [2 - e-'(t +2)] Notes: (i) Clearly show in 'Step 2: Integrating Factor' how P(t) = exp(t)/(1+ t), (ii) Carry out the full calculations in 'Step 4: Verify', and (iii) show all of your work when integrating-by-parts to receive points in 'Step 5: Integrate'. -M = (1 + t) ²e-²¹
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Consider
dM
det
M = (1+ t)²e t
1+ t
subject to the condition M(0) = 0.
(a)
Use the 7-step method to solve this problem and omit 'Step 7:
Sketch'. Clearly label each step and reduce your solution to
M(t) = (1 + t)e¯{[2 – e"(t + 2)]
Notes: (i) Clearly show in 'Step 2: Integrating Factor' how P(t) = exp(t)/(1+
t), (ii) Carry out the full calculations in 'Step 4: Verify', and (iii) show
all of your work when integrating-by-parts to receive points in 'Step 5:
Integrate'.
(b)
Sketch'.
Answer the following questions as part of your completing Step 4:
What is a first approximation to M(t) at large t?
What is a first approximation to M(t) at small t? Use the
Taylor series library exp(small) 1+ small. You should find that
M(t) s t.
) Show that there is at most one zero of the derivative. Do not
find the derivative but only refer to the provided solution in equation
(4) and use logic involving the sign, plus or minus, of various terms in
the solution and the results of (2(b)i) and (2(b)ii).
i.
ii.
iii.
iv.
Use the results of (i), (ii), and (iii) to sketch the solution and
note each of these on the sketch.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88f9be6d-d8dd-4674-ab78-d1dd0cdba8bb%2F0121c108-6c77-41a3-b153-10329583a75e%2Fz4vtbr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider
dM
det
M = (1+ t)²e t
1+ t
subject to the condition M(0) = 0.
(a)
Use the 7-step method to solve this problem and omit 'Step 7:
Sketch'. Clearly label each step and reduce your solution to
M(t) = (1 + t)e¯{[2 – e"(t + 2)]
Notes: (i) Clearly show in 'Step 2: Integrating Factor' how P(t) = exp(t)/(1+
t), (ii) Carry out the full calculations in 'Step 4: Verify', and (iii) show
all of your work when integrating-by-parts to receive points in 'Step 5:
Integrate'.
(b)
Sketch'.
Answer the following questions as part of your completing Step 4:
What is a first approximation to M(t) at large t?
What is a first approximation to M(t) at small t? Use the
Taylor series library exp(small) 1+ small. You should find that
M(t) s t.
) Show that there is at most one zero of the derivative. Do not
find the derivative but only refer to the provided solution in equation
(4) and use logic involving the sign, plus or minus, of various terms in
the solution and the results of (2(b)i) and (2(b)ii).
i.
ii.
iii.
iv.
Use the results of (i), (ii), and (iii) to sketch the solution and
note each of these on the sketch.
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