9ac 30%9.4.5.6.7.8.ProblemsIn each of Problems 1 through 8, solve the given differential equation.x²1. y'==2.3.38 CHAPTER 2 First-Order Differential Equationsyy' + y² sin x = 0y' = cos(x) cos² (2y)xy' = (1 - y2) 1/2x-e-*38 38 3863dydxdydxdydxdydx====1+ y²al%y + ey1-2yy' ==1+3x²Зу2 — буy(0) = 1yIn each of Problems 9 through 16:a. Find the solution of the given initial value problem in explicitform.Gb. Plot the graph of the solution.c. Determine (at least approximately) the interval in which thesolution is defined.y'= (1-2x) y²,y(0) = -1/610.y'=(1-2x)/y, y(1) = −211. xdx+ye*dy = 0, y(0) = 112. dr/d0 = r²/0, r(1) = 213. y'=xy³(1+x²)-1/², y(0) = 1 = (14. y'= 2x/(1+2y), y(2) = 0= (ut.x)\ li seussed brill 23. Solve the equation o15. y' = (3x²-e¹)/(2y-5), y(0) = 1Inilgenotihad to noirulosa aloc= (no16. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3Some of the results requested in Problems 17 through 22 can beobtained either by solving the given equations analytically or byplotting numerically generated approximations to the solutions. Tryto form an opinion about the advantages and disadvantages of eachapproach.150TOG 17. Solve the initial value problemand determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 18. Solve the initial value problem3x²3y² - 4'y' =y(1) = 0ow (TS) brus (OS)G 19.and determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.19. Solve the initial value problemy' = 2y² + xy², y(0) = 1and determine where the solution attains its minimum value.G 20. Solve the initial value problem2- et3+2yy(0) = 0y' ==and determine where the solution attains its maximum value.G21. Consider the initial value problemty(4-y)y' =y(0) = yo.3a. Determine how the behavior of the solution as t increasesdepends on the initial value yo.b. Suppose that yo= 0.5. Find the time T at which the solutionfirst reaches the value 3.98.G 22. Consider the initial value problemdQdtty(4- y)1+ty' =y(0) = yo > 0. Claa.Determine how the solution behaves as t →∞.b. If yo= 2, find the time T at which the solution first reachesthe value 3.99.c. Find the range of initial values for which the solution lies inthe interval 3.99

In this question we have to solve the given differential eq. in the explicit form. Let's solve and…

In each of Problems 9 through 16: a. Find the solution of the given initial value problem in explicit 9. 10. form. Gb. Plot the graph of the solution. c. Determine (at least approximately) the interval in which the solution is defined. y'=(1-2x) y2, y(0) = -1/6 y'=(1-2)( WODY

In each of Problems 9 through 16: a. Find the solution of the given initial value problem in explicit 9. 10. form. Gb. Plot the graph of the solution. c. Determine (at least approximately) the interval in which the solution is defined. y'=(1-2x) y2, y(0) = -1/6 y'=(1-2)( WODY

Database System Concepts

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Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Rod Cutting

A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exponential time complexity. So there is a need to figure out a better method than this.

Question

9ac

30%
9.
4.
5.
6.
7.
8.
Problems
In each of Problems 1 through 8, solve the given differential equation.
x²
1. y'=
=
2.
3.
38 CHAPTER 2 First-Order Differential Equations
y
y' + y² sin x = 0
y' = cos(x) cos² (2y)
xy' = (1 - y2) 1/2
x-e-*
38 38 3863
dy
dx
dy
dx
dy
dx
dy
dx
=
=
=
=
1+ y²
al%
y + ey
1-2
y
y' =
=
1+3x²
Зу2 — бу
y(0) = 1
y
In each of Problems 9 through 16:
a. Find the solution of the given initial value problem in explicit
form.
Gb. Plot the graph of the solution.
c. Determine (at least approximately) the interval in which the
solution is defined.
y'= (1-2x) y²,
y(0) = -1/6
10.
y'=(1-2x)/y, y(1) = −2
11. xdx+ye*dy = 0, y(0) = 1
12. dr/d0 = r²/0, r(1) = 2
13. y'=xy³(1+x²)-1/², y(0) = 1 = (
14. y'= 2x/(1+2y), y(2) = 0
= (ut.x)\ li seussed brill 23. Solve the equation o
15. y' = (3x²-e¹)/(2y-5), y(0) = 1
Inilgenotihad to noirulosa aloc= (no
16. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3
Some of the results requested in Problems 17 through 22 can be
obtained either by solving the given equations analytically or by
plotting numerically generated approximations to the solutions. Try
to form an opinion about the advantages and disadvantages of each
approach.
150
TO
G 17. Solve the initial value problem
and determine the interval in which the solution is valid.
Hint: To find the interval of definition, look for points where the
integral curve has a vertical tangent.
G 18. Solve the initial value problem
3x²
3y² - 4'
y' =
y(1) = 0
ow (TS) brus (OS)
G 19.
and determine the interval in which the solution is valid.
Hint: To find the interval of definition, look for points where the
integral curve has a vertical tangent.
19. Solve the initial value problem
y' = 2y² + xy², y(0) = 1
and determine where the solution attains its minimum value.
G 20. Solve the initial value problem
2- et
3+2y
y(0) = 0
y' =
=
and determine where the solution attains its maximum value.
G21. Consider the initial value problem
ty(4-y)
y' =
y(0) = yo.
3
a. Determine how the behavior of the solution as t increases
depends on the initial value yo.
b. Suppose that yo= 0.5. Find the time T at which the solution
first reaches the value 3.98.
G 22. Consider the initial value problem
dQ
dt
ty(4- y)
1+t
y' =
y(0) = yo > 0. Cla
a.
Determine how the solution behaves as t →∞.
b. If yo= 2, find the time T at which the solution first reaches
the value 3.99.
c. Find the range of initial values for which the solution lies in
the interval 3.99<y< 4.01 by the time t = 2.
2000 dy
dx
ay+bat bastouldy
cy + d'
where a, b, c, and d are constants.
24. Use separation of variables to solve the differential equation
($) moun
pitulce:
=r(a+bQ), Q(0) = 20,
in 188
where a, b, r, and Qo are constants. Determine how the solution
behaves as t→∞
gabrags
=
Homogeneous Equations. If the right-hand side of the equation
dy/dx f(x, y) can be expressed as a function of the ratio y/x
only, then the equation is said to be homogeneous.¹ Such equations
can always be transformed into separable equations by a change of
the dependent variable. Problem 25 illustrates how to solve first-order
homogeneous equations.
USUDS
The word "homogeneous" has different meanings in different mathematical
contexts. The homogeneous equations considered here have nothing to do with
the homogeneous equations that will occur in Chapter 3 and elsewhere.
N

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