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Lecture 3 Exact Equations This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Answer: Exercise 3: Question 1 Let, in the DE of first order and first degree (1.9), the function f(x, y) = -M(x,y)/N(x,y), so that it can be written as M(x, y) + N(x,y)y ------ 口 0, (3.1) where M and N are continuous functions having continuous partial deriva- tives My and N, in the rectangle S: 2-2o|

Lecture 3 Exact Equations This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Answer: Exercise 3: Question 1 Let, in the DE of first order and first degree (1.9), the function f(x, y) = -M(x,y)/N(x,y), so that it can be written as M(x, y) + N(x,y)y ------ 口 0, (3.1) where M and N are continuous functions having continuous partial deriva- tives My and N, in the rectangle S: 2-2o|

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 44E
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Lecture 3 Exact Equations This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Answer: Exercise 3: Question 1 Let, in the DE of first order and first degree (1.9), the function f(x, y) = -M(x,y)/N(x,y), so that it can be written as M(x, y) + N(x,y)y ------ 口 0, (3.1) where M and N are continuous functions having continuous partial deriva- tives My and N, in the rectangle S: 2-2o|<a, ly-30<b (0<a, b< ∞). The nomenclature comes from the fact that Equation (3.1) is said to be exact if there exists a function u(x, y) such that (x,y) M(x,y) and (x,y) = N(x,y). (3.2) M+Ny' = U₂+ Uyy is exactly the derivative du/dr. Once the DE (3.1) is exact its implicit solution is u(x,y) = c. Theorem 3.1. Let the functions M(x, y) and N(x, y) together with their partial derivatives M,(x, y) and N2(x, y) be continuous in the rect- angle Sr-ol<a, yo<b (0<a, b< ∞). Then the DE (3.1) is exact if and only if condition (3.4) is satisfied. Obviously, in this result S may be replaced by any region which does not include any "hole." The above proof of this theorem is, in fact, constructive, i.e., we can explicitly find a solution of (3.1). For this, we compute g(y) from (3.6), +M(8, 1)ds + 9(10). Yo (u) = N(x, t)dt - M(s,y)ds + Therefore, from (3.5) it follows that u(x,y) = N(r,t)dt + M(s,yo)ds+9(30) Mo and hence a solution of the exact equation (3.1) is given by "N(r,t)dt+M(s, 0)ds = c₁ where e is an arbitrary constant. (3.3) (3.7)
Transcribed Image Text:Lecture 3 Exact Equations This is the reference book, please answer the following question. Ravi P. Agarwal Donal O'Regan An Introduction to Ordinary Differential Equations Answer: Exercise 3: Question 1 Let, in the DE of first order and first degree (1.9), the function f(x, y) = -M(x,y)/N(x,y), so that it can be written as M(x, y) + N(x,y)y ------ 口 0, (3.1) where M and N are continuous functions having continuous partial deriva- tives My and N, in the rectangle S: 2-2o|<a, ly-30<b (0<a, b< ∞). The nomenclature comes from the fact that Equation (3.1) is said to be exact if there exists a function u(x, y) such that (x,y) M(x,y) and (x,y) = N(x,y). (3.2) M+Ny' = U₂+ Uyy is exactly the derivative du/dr. Once the DE (3.1) is exact its implicit solution is u(x,y) = c. Theorem 3.1. Let the functions M(x, y) and N(x, y) together with their partial derivatives M,(x, y) and N2(x, y) be continuous in the rect- angle Sr-ol<a, yo<b (0<a, b< ∞). Then the DE (3.1) is exact if and only if condition (3.4) is satisfied. Obviously, in this result S may be replaced by any region which does not include any "hole." The above proof of this theorem is, in fact, constructive, i.e., we can explicitly find a solution of (3.1). For this, we compute g(y) from (3.6), +M(8, 1)ds + 9(10). Yo (u) = N(x, t)dt - M(s,y)ds + Therefore, from (3.5) it follows that u(x,y) = N(r,t)dt + M(s,yo)ds+9(30) Mo and hence a solution of the exact equation (3.1) is given by "N(r,t)dt+M(s, 0)ds = c₁ where e is an arbitrary constant. (3.3) (3.7)
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