This question is about the differential equation dy Y = t³(y - t)² + dt t For each of the following statements about this differential equation, determine if the state- ment is true or false and give your reasons. (a) y(t) = t - is a solution to the differential equation. (b) The Existence Theorem implies that the initial value problem consisting of the dif- ferential equation and the initial condition y(1) = 1 has a solution that exists for all values of t> 0. (c) The Uniqueness Theorem implies that y(t) = t is the only solution to the initial value problem consisting of the differential equation along with the initial condition y (0) = 0. (d) Using Euler's method on the initial value problem in (b), with a stepsize of h = 1, yields the approximation y(3)≈ 4.
This question is about the differential equation dy Y = t³(y - t)² + dt t For each of the following statements about this differential equation, determine if the state- ment is true or false and give your reasons. (a) y(t) = t - is a solution to the differential equation. (b) The Existence Theorem implies that the initial value problem consisting of the dif- ferential equation and the initial condition y(1) = 1 has a solution that exists for all values of t> 0. (c) The Uniqueness Theorem implies that y(t) = t is the only solution to the initial value problem consisting of the differential equation along with the initial condition y (0) = 0. (d) Using Euler's method on the initial value problem in (b), with a stepsize of h = 1, yields the approximation y(3)≈ 4.
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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Transcribed Image Text:This question is about the differential equation
dy = t³ (y − 1)² + 1/1/
dt
For each of the following statements about this differential equation, determine if the state-
ment is true or false and give your reasons.
(a) y(t) = t - is a solution to the differential equation.
(b) The Existence Theorem implies that the initial value problem consisting of the dif-
ferential equation and the initial condition y(1) = 1 has a solution that exists for all
values of t> 0.
(c) The Uniqueness Theorem implies that y(t) = t is the only solution to the initial
value problem consisting of the differential equation along with the initial condition
y (0) = 0.
(d) Using Euler's method on the initial value problem in (b), with a stepsize of h = 1,
yields the approximation y(3) ≈ 4.
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