This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Please answer Question 1 from chapter 4 of above mentioned book, all references are attached. Lecture 4 Elementary First-Order Equations Suppose in the DE of first order (3.1). M(r,y) = X₁(r), (y) and N()-X2()(), so that it takes the form X()() | X()2(n) = 0. (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(z) Y₂(v) X2()() + = 0 (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 -de + Y₁(w) (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 (2) X₂() 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). Example 4.1. The DE in Example 3.2 may be written as 1 1 = 0, ry(1-3) 0 y(1-y) for which (4.3) gives the solution y = (1-er)-1. Other possible solutions for which (-2)=0 are r = 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-cr). 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 f(x,y) = *(3) (4.4) For example, the functions 32-ry-y², sin(x²/(x²- y²)), (x² + 7y)1/5, (1/2) sin(x/y)+(x/y3)(Iny-Inz), and (6e//r2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if A 1/2, then it is the same as f(x,y). (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/x).
This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Please answer Question 1 from chapter 4 of above mentioned book, all references are attached. Lecture 4 Elementary First-Order Equations Suppose in the DE of first order (3.1). M(r,y) = X₁(r), (y) and N()-X2()(), so that it takes the form X()() | X()2(n) = 0. (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(z) Y₂(v) X2()() + = 0 (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 -de + Y₁(w) (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 (2) X₂() 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). Example 4.1. The DE in Example 3.2 may be written as 1 1 = 0, ry(1-3) 0 y(1-y) for which (4.3) gives the solution y = (1-er)-1. Other possible solutions for which (-2)=0 are r = 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-cr). 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 f(x,y) = *(3) (4.4) For example, the functions 32-ry-y², sin(x²/(x²- y²)), (x² + 7y)1/5, (1/2) sin(x/y)+(x/y3)(Iny-Inz), and (6e//r2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if A 1/2, then it is the same as f(x,y). (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/x).
Algebra & Trigonometry with Analytic Geometry
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ISBN:9781133382119
Author:Swokowski
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Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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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
Please answer Question 1 from
chapter 4 of above mentioned
book, all references are attached.
Lecture 4
Elementary First-Order
Equations
Suppose in the DE of first order (3.1). M(r,y) = X₁(r), (y) and
N()-X2()(), so that it takes the form
X()() | X()2(n) = 0.
(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(z) Y₂(v)
X2()()
+
= 0
(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
-de +
Y₁(w)
(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 (2) X₂()
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).
Example 4.1. The DE in Example 3.2 may be written as
1
1
= 0, ry(1-3) 0
y(1-y)
for which (4.3) gives the solution y = (1-er)-1. Other possible solutions for
which (-2)=0 are r = 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-cr).
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
f(x,y) = *(3)
(4.4)
For example, the functions 32-ry-y², sin(x²/(x²- y²)), (x² +
7y)1/5, (1/2) sin(x/y)+(x/y3)(Iny-Inz), and (6e//r2/31/3) are ho-
mogeneous of degree 2, 0, 4/5, -2, and -1, respectively.
In (4.4) if A 1/2, then it is the same as
f(x,y).
(4.5)
This implies that a homogeneous function of degree zero is a function of a
single variable v (=y/x).
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