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Consider the differential equation (D₁) : 2ċ(t) — 8e¯ä(t) +5= 0, x(0) = ln(4) and the relation y(t) = eä(t) — 3. 1. Show that y(t) satisfies the differential equation (D₂) : 2ÿ(t)+5y(t)+7= 0, y(0) = 1 if and only if x(t) satisfies the differential equation (D₁).

Consider the differential equation (D₁) : 2ċ(t) — 8e¯ä(t) +5= 0, x(0) = ln(4) and the relation y(t) = eä(t) — 3. 1. Show that y(t) satisfies the differential equation (D₂) : 2ÿ(t)+5y(t)+7= 0, y(0) = 1 if and only if x(t) satisfies the differential equation (D₁).

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) : 2x(t) — 8e¯ï(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 x(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.
Transcribed Image Text:Consider the differential equation (D₁) : 2x(t) — 8e¯ï(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 x(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.
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