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(x,y) N() X2()2(y), so that it takes the form X₁(r)i(y) and X₁()() | X()Y₂(n) = (1. (4.1) If Y₁(y)X(r) 0 for all (r, y) e S, then (4.1) can be written as an exact DE X₁(r) (u) + X2(x) Y₁() = 0 (4.2) Example 4.1. The DE in Example 3.2 may be written as 1 1 Iy(1-y) 0, zy(1-y) 0 for which (4.3) gives the solution y = (1-cr). Other possible solutions for which r(y-y²)=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 r=0 is not a solution, and hence all solutions of this DE are given by y = 0, y = (1-cz)-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 Xf(,). (Ax, Ag) A(2.9). (4.4) Please answer Question 4 from chapter 4 of above mentioned book, all references are attached. 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 X2(x) Y(y) dy Y₁(M) (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 (3) 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 32- -xy-2, sin(2/(x² - y²)), (x² + 7y)1/5, (1/2) sin(x/y)+(x/y³)(lny-Inz), and (6e//r2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if=1/x, then it is the same as **1(1)-1(x) = f(x,y). (4.5) This implies that a homogeneous function of degree zero is a function of a single variable (=y/x).
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(x,y) N() X2()2(y), so that it takes the form X₁(r)i(y) and X₁()() | X()Y₂(n) = (1. (4.1) If Y₁(y)X(r) 0 for all (r, y) e S, then (4.1) can be written as an exact DE X₁(r) (u) + X2(x) Y₁() = 0 (4.2) Example 4.1. The DE in Example 3.2 may be written as 1 1 Iy(1-y) 0, zy(1-y) 0 for which (4.3) gives the solution y = (1-cr). Other possible solutions for which r(y-y²)=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 r=0 is not a solution, and hence all solutions of this DE are given by y = 0, y = (1-cz)-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 Xf(,). (Ax, Ag) A(2.9). (4.4) Please answer Question 4 from chapter 4 of above mentioned book, all references are attached. 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 X2(x) Y(y) dy Y₁(M) (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 (3) 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 32- -xy-2, sin(2/(x² - y²)), (x² + 7y)1/5, (1/2) sin(x/y)+(x/y³)(lny-Inz), and (6e//r2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if=1/x, then it is the same as **1(1)-1(x) = f(x,y). (4.5) This implies that a homogeneous function of degree zero is a function of a single variable (=y/x).
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
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(x,y)
N() X2()2(y), so that it takes the form
X₁(r)i(y) and
X₁()() | X()Y₂(n) = (1.
(4.1)
If Y₁(y)X(r) 0 for all (r, y) e S, then (4.1) can be written as an exact DE
X₁(r) (u)
+
X2(x) Y₁()
= 0
(4.2)
Example 4.1. The DE in Example 3.2 may be written as
1
1
Iy(1-y)
0, zy(1-y) 0
for which (4.3) gives the solution y = (1-cr). Other possible solutions for
which r(y-y²)=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 r=0 is not a solution,
and hence all solutions of this DE are given by y = 0, y = (1-cz)-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
Xf(,).
(Ax, Ag) A(2.9).
(4.4)
Please answer Question 4 from
chapter 4 of above mentioned
book, all references are attached.
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
X2(x)
Y(y)
dy
Y₁(M)
(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 (3) 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 32- -xy-2, sin(2/(x² - y²)), (x² +
7y)1/5, (1/2) sin(x/y)+(x/y³)(lny-Inz), and (6e//r2/31/3) are ho-
mogeneous of degree 2, 0, 4/5, -2, and -1, respectively.
In (4.4) if=1/x, then it is the same as
**1(1)-1(x)
=
f(x,y).
(4.5)
This implies that a homogeneous function of degree zero is a function of a
single variable (=y/x).
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