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 the complex roots. r = α ± i β (3) Write the expression for general solution of a y ″ + b y ′ + c y = 0 with complex roots. y = e α x ( c 1 cos β x + c 2 sin β x ) (4) Here, α is the real part of the root, and β is the imaginary part of the root. Consider the differential equation as follows. d 2 y d x 2 + 2 d y d x + 2 y = 0 (5) Compare equation (1) and (5). a = 1 b = 2 c = 2 Find the auxiliary equation. Substitute 1 for a , 2 for b and 2 for c in equation (2), ( 1 ) r 2 + ( 2 ) r + ( 2 ) = 0 r 2 + 2 r + 2 = 0 Solve for r . r = − ( 2 ) ± ( 2 ) 2 − 4 ( 1 ) ( 2 ) 2 ( 1 ) = − ( 2 ) ± 4 − 8 2 = − 2 ± − 4 2 = − 2 ± 2 i 2 Simplify r as follows. r = − 1 ± i (6) Compare equations (3) and (6). α = − 1 β = 1 Substitute − 1 for α and 1 for β in equation (4), y = e ( − 1 ) x ( c 1 cos ( 1 ) x + c 2 sin ( 1 ) x ) y = e − x ( c 1 cos x + c 2 sin x ) (7) Here, c 1 and c 2 are constant value. Consider the value of c 1 and c 2 as 1 . Substitute 1 for c 1 and 1 for c 2 in equation (7), y = e − x ( ( 1 ) cos x + ( 1 ) sin x ) = e − x ( cos x + sin x ) y = e − x cos x + e − x sin x (8) Equation (8) consists two functions f ( x ) and g ( x ) as follows

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Author: James Stewart

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Differential Equations

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Chapter 17.1, Problem 15E

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To graph: The two basic solutions along with several other solutions of the differential equation for d2ydx2+2dydx+2y=0 .

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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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To graph: The two basic solutions along with several other solutions of the differential equation for d 2 y d x 2 + 2 d y d x + 2 y = 0 .

Solution Summary: The author explains how the graph of two basic solutions along with several other solutions of the differential equation is drawn.

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To graph: The two basic solutions along with several other solutions of the differential equation for d2ydx2+2dydx+2y=0 .

Chapter 17 Solutions