This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Lecture 4 Elementary First-Order Equations Suppose in the DE of first order (3.1), M(r,y) N(,) X()(), so that it takes the form X₁(r)i(y) and X₁()() | X()2() = 0). (4.1) If Y₁(y)X()0 for all (r,y) e S, then (4.1) can be written as an exact DE X₁(x) (y) Xa(z) Yi() + (4.2) in which the variables are separated. Such a DE (4.2) is said to be separable. The solution of this exact equation is given by Example 4.1. The DE in Example 3.2 may be written as T 1 y(1-y) 0, ry(1-y) 0 for which (4.3) gives the solution y = (1-x)-1. Other possible solutions for which (-2)=0 are x = 0, y = 0, and y = 1. However, the solution y = 1 is already included in y = (1-cr) for c=0, and z = 0 is not a solution, and hence all solutions of this DE are given by y = 0, y = (1-cr)¹ A function f(x, y) defined in a domain D (an open connected set in IR²) is said to be homogeneous of degree k if for all real A and (x, y) = D (Xxx, Ay) = (2.). (4.4) Please answer Question 5 from chapter 4 of above mentioned book, all references are attached. - X₂(z) +/- Yz(y) Y₁(w) dy G (1.3) Here both the integrals are indefinite and constants of integration have been absorbed in c. Equation (4.3) contains all the solutions of (4.1) for which Yi (7) X2(x) 0. In fact, when we divide (1.1) by YX2 we might have lost some solutions, and the ones which are not in (4.3) for some e must be coupled with (4.3) to obtain all solutions of (4.1). For example, the functions 3x²-ry-y², sin(x²/(x2 y2)), (x4 + 7y1)1/5, (1/2) sin(x/y)+(x/y³) (Iny-Inr), and (6e//r2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if X 1/x, then it is the same as xf (1,2) = f(x,y). (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/).
This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Lecture 4 Elementary First-Order Equations Suppose in the DE of first order (3.1), M(r,y) N(,) X()(), so that it takes the form X₁(r)i(y) and X₁()() | X()2() = 0). (4.1) If Y₁(y)X()0 for all (r,y) e S, then (4.1) can be written as an exact DE X₁(x) (y) Xa(z) Yi() + (4.2) in which the variables are separated. Such a DE (4.2) is said to be separable. The solution of this exact equation is given by Example 4.1. The DE in Example 3.2 may be written as T 1 y(1-y) 0, ry(1-y) 0 for which (4.3) gives the solution y = (1-x)-1. Other possible solutions for which (-2)=0 are x = 0, y = 0, and y = 1. However, the solution y = 1 is already included in y = (1-cr) for c=0, and z = 0 is not a solution, and hence all solutions of this DE are given by y = 0, y = (1-cr)¹ A function f(x, y) defined in a domain D (an open connected set in IR²) is said to be homogeneous of degree k if for all real A and (x, y) = D (Xxx, Ay) = (2.). (4.4) Please answer Question 5 from chapter 4 of above mentioned book, all references are attached. - X₂(z) +/- Yz(y) Y₁(w) dy G (1.3) Here both the integrals are indefinite and constants of integration have been absorbed in c. Equation (4.3) contains all the solutions of (4.1) for which Yi (7) X2(x) 0. In fact, when we divide (1.1) by YX2 we might have lost some solutions, and the ones which are not in (4.3) for some e must be coupled with (4.3) to obtain all solutions of (4.1). For example, the functions 3x²-ry-y², sin(x²/(x2 y2)), (x4 + 7y1)1/5, (1/2) sin(x/y)+(x/y³) (Iny-Inr), and (6e//r2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if X 1/x, then it is the same as xf (1,2) = f(x,y). (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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Transcribed Image Text:This is the reference book, please
answer the following question.
Ravi P. Agarwal
Donal O'Regan
An Introduction to
Ordinary Differential
Equations
Lecture 4
Elementary First-Order
Equations
Suppose in the DE of first order (3.1), M(r,y)
N(,) X()(), so that it takes the form
X₁(r)i(y) and
X₁()() | X()2() = 0).
(4.1)
If Y₁(y)X()0 for all (r,y) e S, then (4.1) can be written as an exact DE
X₁(x) (y)
Xa(z) Yi()
+
(4.2)
in which the variables are separated. Such a DE (4.2) is said to be separable.
The solution of this exact equation is given by
Example 4.1. The DE in Example 3.2 may be written as
T
1
y(1-y)
0, ry(1-y) 0
for which (4.3) gives the solution y = (1-x)-1. Other possible solutions for
which (-2)=0 are x = 0, y = 0, and y = 1. However, the solution y = 1
is already included in y = (1-cr) for c=0, and z = 0 is not a solution,
and hence all solutions of this DE are given by y = 0, y = (1-cr)¹
A function f(x, y) defined in a domain D (an open connected set in IR²)
is said to be homogeneous of degree k if for all real A and (x, y) = D
(Xxx, Ay)
=
(2.).
(4.4)
Please answer Question 5 from
chapter 4 of above mentioned
book, all references are attached.
-
X₂(z)
+/-
Yz(y)
Y₁(w)
dy
G
(1.3)
Here both the integrals are indefinite and constants of integration have been
absorbed in c.
Equation (4.3) contains all the solutions of (4.1) for which Yi (7) X2(x)
0. In fact, when we divide (1.1) by YX2 we might have lost some solutions,
and the ones which are not in (4.3) for some e must be coupled with (4.3)
to obtain all solutions of (4.1).
For example, the functions 3x²-ry-y², sin(x²/(x2 y2)), (x4 +
7y1)1/5, (1/2) sin(x/y)+(x/y³) (Iny-Inr), and (6e//r2/31/3) are ho-
mogeneous of degree 2, 0, 4/5, -2, and -1, respectively.
In (4.4) if X 1/x, then it is the same as
xf (1,2) = f(x,y).
(4.5)
This implies that a homogeneous function of degree zero is a function of a
single variable v (=y/).
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