21 the initial value problem= 2 cosy' + =y=Find the coordinates of the first local maximum point of the solutionfor t > 0.N 18.Consider the initial value problemy' +y=1-1/4, y(0) = yo.the t-axis.Find the value of yo for which the solution touches, but does not cross,19. Consider the initial value problem2 cost, y(0) = -1.1y + y = 3 + 2 cos(2t), y(0) = 0.a. Find the solution of this initial value problem and describe itsbehavior for large t.N b. Determine the value of t for which the solution firstintersects the line y = 12.20. Find the value of yo for which the solution of the initial valueproblem1000JONESy'y' - y = 1 + 3 sint, y(0) = yoremains finite as t → ∞.21. Consider the initial value problem-3y = 3t+2e', y(0) = Yo.Find the value of yo that separates solutions that grow positively ast → ∞ from those that grow negatively. How does the solution thatcorresponds to this critical value of yo behave as t → ∞?22. Show that all solutions of 2y' + ty = 2 [equation (41) of thetext] approach a limit as t→∞, and find the limiting value.Hint: Consider the general solution, equation (47). Show that the first25.All solut26.All solut27. All solut28. Variatiosolving the gera. If g(twhere Ab. If g(equationwhere Adifferentic. Findequationin this mtechniquit is discuorder linIn each of Prothe given diff29. y'-2y30. y'+y² + 1/3

The initial value problem y1-32y=3t+2et, y0=y0Linear differential first order equation,Initial…

21. Consider the initial value problem 3 y' = 2y = 3t+2e², 3t+2e', y(0) = Yo. Find the value of yo that separates solutions that grow positively as t→∞ from those that grow negatively. How does the solution that corresponds to this critical value of yo behave as t → ∞o?

21. Consider the initial value problem 3 y' = 2y = 3t+2e², 3t+2e', y(0) = Yo. Find the value of yo that separates solutions that grow positively as t→∞ from those that grow negatively. How does the solution that corresponds to this critical value of yo behave as t → ∞o?

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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Question

100%

the initial value problem
= 2 cos
y' + =y=
Find the coordinates of the first local maximum point of the solution
for t > 0.
N 18.
Consider the initial value problem
y' +
y=1-1/4, y(0) = yo.
the t-axis.
Find the value of yo for which the solution touches, but does not cross,
19. Consider the initial value problem
2 cost, y(0) = -1.
1
y + y = 3 + 2 cos(2t), y(0) = 0.
a. Find the solution of this initial value problem and describe its
behavior for large t.
N b. Determine the value of t for which the solution first
intersects the line y = 12.
20. Find the value of yo for which the solution of the initial value
problem
1000
JONES
y'
y' - y = 1 + 3 sint, y(0) = yo
remains finite as t → ∞.
21. Consider the initial value problem
-
3
y = 3t+2e', y(0) = Yo.
Find the value of yo that separates solutions that grow positively as
t → ∞ from those that grow negatively. How does the solution that
corresponds to this critical value of yo behave as t → ∞?
22. Show that all solutions of 2y' + ty = 2 [equation (41) of the
text] approach a limit as t→∞, and find the limiting value.
Hint: Consider the general solution, equation (47). Show that the first
25.
All solut
26.
All solut
27. All solut
28. Variatio
solving the ger
a. If g(t
where A
b. If g(
equation
where A
differenti
c. Find
equation
in this m
techniqu
it is discu
order lin
In each of Pro
the given diff
29. y'-2y
30. y'+
y² + 1/3

Expert Solution

Step 1

$T h e i n i t i a l v a l u e p r o b l e m y^{1} - \frac{3}{2} y = 3 t + 2 e^{t}, y (0) = y_{0} L i n e a r d i f f e r e n t i a l f i r s t o r d e r e q u a t i o n, I n i t i a l f u n c t i o n = e^{\int \frac{- 3}{2} d t} = e^{\frac{- 3 t}{2}} s o l u t i o n i s g i v e n b y y e^{\frac{- 3 t}{2}} = \int (3 t + 2 e^{t}) e^{\frac{- 3 t}{2}} d t + c y e^{\frac{- 3 t}{2}} = \int 3 t e^{\frac{- 3 t}{2}} d t + 2 \int e^{\frac{- t}{2} d t} + c$