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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 3 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)i(y) and N(x,y) X2()2(v), so that it takes the form X()() | X()() = 11. (4.1) If Y₁(3)X(r) 0 for all (r,y) e S, then (4.1) can be written as an exact DE X₁(x)Y(y) + Xa(z) ()=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 · X₁(2), X2(e) dx+ +/- Y(y). dy Y₁(3) (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 (1) X2(1) 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-y) 0 y(1-y) for which (4.3) gives the solution y = (1-cr). Other possible solutions for which (y-3)=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 R²) is said to be homogeneous of degree k if for all real A and (x, y) = D (A, Ay) (2,9). Xf(x,y). (4.4) For example, the functions 32-xy-y², sin(x²/(2 y2)), (x4 + 7y)1/5, (1/2) sin(x/y)+(x/y3)(lny-In), and (6e//2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if 1/2, then it is the same as *1(1)-(TN) (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/1).

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 3 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)i(y) and N(x,y) X2()2(v), so that it takes the form X()() | X()() = 11. (4.1) If Y₁(3)X(r) 0 for all (r,y) e S, then (4.1) can be written as an exact DE X₁(x)Y(y) + Xa(z) ()=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 · X₁(2), X2(e) dx+ +/- Y(y). dy Y₁(3) (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 (1) X2(1) 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-y) 0 y(1-y) for which (4.3) gives the solution y = (1-cr). Other possible solutions for which (y-3)=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 R²) is said to be homogeneous of degree k if for all real A and (x, y) = D (A, Ay) (2,9). Xf(x,y). (4.4) For example, the functions 32-xy-y², sin(x²/(2 y2)), (x4 + 7y)1/5, (1/2) sin(x/y)+(x/y3)(lny-In), and (6e//2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if 1/2, then it is the same as *1(1)-(TN) (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/1).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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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 3 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)i(y) and N(x,y) X2()2(v), so that it takes the form X()() | X()() = 11. (4.1) If Y₁(3)X(r) 0 for all (r,y) e S, then (4.1) can be written as an exact DE X₁(x)Y(y) + Xa(z) ()=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 · X₁(2), X2(e) dx+ +/- Y(y). dy Y₁(3) (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 (1) X2(1) 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-y) 0 y(1-y) for which (4.3) gives the solution y = (1-cr). Other possible solutions for which (y-3)=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 R²) is said to be homogeneous of degree k if for all real A and (x, y) = D (A, Ay) (2,9). Xf(x,y). (4.4) For example, the functions 32-xy-y², sin(x²/(2 y2)), (x4 + 7y)1/5, (1/2) sin(x/y)+(x/y3)(lny-In), and (6e//2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if 1/2, then it is the same as *1(1)-(TN) (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/1).
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 3 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)i(y) and N(x,y) X2()2(v), so that it takes the form X()() | X()() = 11. (4.1) If Y₁(3)X(r) 0 for all (r,y) e S, then (4.1) can be written as an exact DE X₁(x)Y(y) + Xa(z) ()=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 · X₁(2), X2(e) dx+ +/- Y(y). dy Y₁(3) (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 (1) X2(1) 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-y) 0 y(1-y) for which (4.3) gives the solution y = (1-cr). Other possible solutions for which (y-3)=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 R²) is said to be homogeneous of degree k if for all real A and (x, y) = D (A, Ay) (2,9). Xf(x,y). (4.4) For example, the functions 32-xy-y², sin(x²/(2 y2)), (x4 + 7y)1/5, (1/2) sin(x/y)+(x/y3)(lny-In), and (6e//2/31/3) are ho- mogeneous of degree 2, 0, 4/5, -2, and -1, respectively. In (4.4) if 1/2, then it is the same as *1(1)-(TN) (4.5) This implies that a homogeneous function of degree zero is a function of a single variable v (=y/1).
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