C. (a) State the uniqueness and existence theorem for the initial value problem dy -=f(y,t), y(t) = yo. dt Consider the initial value problem dy -1)- +3y=2, y(t) = Yo dt Determine all pairs (yo,to) for which the uniqueness of the solution is not guaranteed.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2.
(a) State the uniqueness and existence theorem for the initial value problem
dy
=f(y,t), y(t) = yo.
dt
Consider the initial value problem
dy
(t-1) +3y=2, y(t) = yo
dt
Determine all pairs (yo,to) for which the uniqueness of the solution is not
guaranteed.
1
(b) Verify that y₁ (t)=1 and y₂(t)=t² are two solutions of the differential
equation yy"+(y)² = 0 for t>0. Then show that
C₁+C₂t
is not, in general, a solution of this equation. Explain why this result does not
contradict the Principle of Superposition.
Transcribed Image Text:2. (a) State the uniqueness and existence theorem for the initial value problem dy =f(y,t), y(t) = yo. dt Consider the initial value problem dy (t-1) +3y=2, y(t) = yo dt Determine all pairs (yo,to) for which the uniqueness of the solution is not guaranteed. 1 (b) Verify that y₁ (t)=1 and y₂(t)=t² are two solutions of the differential equation yy"+(y)² = 0 for t>0. Then show that C₁+C₂t is not, in general, a solution of this equation. Explain why this result does not contradict the Principle of Superposition.
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