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Consider the differential equation (D₁): 2i(t) — 8e-z(t) +5=0, x(0) = ln(4) and the relation y(t) = e(t) - 3. 1. Show that y(t) satisfies the differential equation (D₂): 2y(t)+5y(t)+7= 0, y(0) = 1 if and only if r(t) satisfies the differential equation (D₁). 2. Find the backward solution of (D₂) and deduce the backward solution of (D₁). 3. Is the solution of (D₁) convergent or divergent? Justify your answer.

Consider the differential equation (D₁): 2i(t) — 8e-z(t) +5=0, x(0) = ln(4) and the relation y(t) = e(t) - 3. 1. Show that y(t) satisfies the differential equation (D₂): 2y(t)+5y(t)+7= 0, y(0) = 1 if and only if r(t) satisfies the differential equation (D₁). 2. Find the backward solution of (D₂) and deduce the backward solution of (D₁). 3. Is the solution of (D₁) convergent or divergent? Justify your answer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the differential equation (D₁): 2i(t)- 8e-z(t) +5= 0, r(0) = ln(4) and the relation y(t) = e(t) - 3. 1. Show that y(t) satisfies the differential equation (D₂): 2y(t)+5y(t)+7= 0, y(0) = 1 if and only if r(t) satisfies the differential equation (D₁). 2. Find the backward solution of (D₂) and deduce the backward solution of (D₁). 3. Is the solution of (D₁) convergent or divergent? Justify your answer. 4. Determine the stationary solution of (D₁) and indicate whether it is stable or unstable. 5. Sketch a phase diagram and a time-path diagram of (D₁). For questions 4 and 5, you can exploit Figure 1 below. 1 3 ff(x) 2 3 10 02 5 co (C) Figure 1: The graph (C) of the function f defined on R by f(x) = 4e¯² — 1/₁
Transcribed Image Text:Consider the differential equation (D₁): 2i(t)- 8e-z(t) +5= 0, r(0) = ln(4) and the relation y(t) = e(t) - 3. 1. Show that y(t) satisfies the differential equation (D₂): 2y(t)+5y(t)+7= 0, y(0) = 1 if and only if r(t) satisfies the differential equation (D₁). 2. Find the backward solution of (D₂) and deduce the backward solution of (D₁). 3. Is the solution of (D₁) convergent or divergent? Justify your answer. 4. Determine the stationary solution of (D₁) and indicate whether it is stable or unstable. 5. Sketch a phase diagram and a time-path diagram of (D₁). For questions 4 and 5, you can exploit Figure 1 below. 1 3 ff(x) 2 3 10 02 5 co (C) Figure 1: The graph (C) of the function f defined on R by f(x) = 4e¯² — 1/₁
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