Explanation Formula used: Write the expression for differential equation. a y ″ + b y ′ + c y = 0 (1) Write the expression for auxiliary equation. a r 2 + b r + c = 0 (2) Write the expression for general solution of a y ″ + b y ′ + c y = 0 with two distinct real roots ( r ) . y = c 1 e r 1 x + c 2 e r 2 x (3) Here, r 1 and r 2 is the root of auxiliary equation. Consider the differential equation as follows. y ″ = y ′ y ″ − y ′ = 0 (4) Compare equation (1) and (4). a = 1 b = − 1 c = 0 Find the auxiliary equation. Substitute 1 for a , − 1 for b , and 0 for c in equation (2), ( 1 ) r 2 + ( − 1 ) r + ( 0 ) = 0 r 2 − r = 0 r ( r − 1 ) = 0 Solve for r . r = 0 r = 1 Consider the value of r 1 and r 2 as follows. r 1 = 0 r 2 = 1 Find the general solution of y ″ = y ′ using equation (3). Substitute 0 for r 1 and 1 for r 2 in equation (3), y = c 1 e ( 0 ) x + c 2 e ( 1 ) x y = c 1 + c 2 e x (5) Modify equation (5) as follows. y ( x ) = c 1 + c 2 e x (6) Find the value of y ( 0 ) . Substitute 0 for x in equation (6), y ( 0 ) = c 1 + c 2 e 0 = c 1 + c 2 Substitute 1 for y ( 0 ) , 1 = c 1 + c 2 (7) Find the value of y ( 1 )

8th Edition

ISBN: 9781305266643

Author: James Stewart

Publisher: Cengage Learning

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Chapter 17.1, Problem 29E

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To solve: The boundary-value problem for differential equation y″=y′ , y(0)=1 , y(1)=2 .

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Chapter 17.1, Problem 28E

Chapter 17.1, Problem 30E

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Chapter 17 Solutions

Multivariable Calculus

Ch. 17.1 - Solve the differential equation. 1. y" y' 6y = 0 Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Solve the differential equation. 4. y" + y' 12y =...Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Solve the differential equation. 9. y" 4y' + 13y...Ch. 17.1 - Prob. 10E

Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Solve the initial-value problem. 19. 9y" + 12y' +...Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Solve the initial-value problem. 23. y" y' 12y =...Ch. 17.1 - Solve the initial-value problem. 24. 4y" + 4y' +...Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Solve the boundary-value problem, if possible. 32....Ch. 17.1 - Prob. 33E Ch. 17.1 - If a, b, and c are all positive constants and y(x)...Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Solve the differential equation using (a)...Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.3 - A spring has natural length 0.75 m and a 5-kg...Ch. 17.3 - Prob. 2E Ch. 17.3 - A spring with a mass of 2 kg has damping constant...Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Suppose a spring has mass m and spring constant k...Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - The battery in Exercise 14 is replaced by a...Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.4 - Prob. 1E Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17 - (a) Write the general form of a second-order...Ch. 17 - Prob. 2RCC Ch. 17 - (a) Write the general form of a second-order...Ch. 17 - Prob. 4RCC Ch. 17 - Prob. 5RCC Ch. 17 - Prob. 1RQ Ch. 17 - Prob. 2RQ Ch. 17 - Prob. 3RQ Ch. 17 - Prob. 4RQ Ch. 17 - Solve the differential equation. 1. 4y" y =0 Ch. 17 - Prob. 2RE Ch. 17 - Prob. 3RE Ch. 17 - Solve the differential equation. 4. y" + 8y' + 16y...Ch. 17 - Prob. 5RE Ch. 17 - Prob. 6RE Ch. 17 - Prob. 7RE Ch. 17 - Prob. 8RE Ch. 17 - Prob. 9RE Ch. 17 - Solve the differential equation. 10....Ch. 17 - Prob. 11RE Ch. 17 - Solve the initial-value problem. 12. y" 6y' + 25y...Ch. 17 - Solve the initial-value problem. 13. y" 5y' + 4y...Ch. 17 - Prob. 14RE Ch. 17 - Solve the boundary-value problem, if possible. 15....Ch. 17 - Prob. 16RE Ch. 17 - Prob. 17RE Ch. 17 - Prob. 18RE Ch. 17 - A series circuit contains a resistor with R = 40 ,...Ch. 17 - Prob. 20RE Ch. 17 - Assume that the earth is a solid sphere of uniform...

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The boundary-value problem for differential equation y ″ = y ′ , y ( 0 ) = 1 , y ( 1 ) = 2 .

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To solve: The boundary-value problem for differential equation y″=y′ , y(0)=1 , y(1)=2 .

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Chapter 17 Solutions