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Math Advanced Math Fundamentals of Differential Equations (9th Edition)

The general solution to the differential equation y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x . The general solution to the differential equation y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x is, y ( x ) = e x + x 2 + c 1 e − 2 x + c 2 e − 3 x _ . Given that, the particular solution to y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x is y p ( x ) = e x + x 2 . The associated auxiliary equation is, r 2 + 5 r + 6 r = 0 . Simplify r 2 + 5 r + 6 r = 0 as, r 2 + 2 r + 3 r + 6 r = 0 r ( r + 2 ) + 3 ( r + 2 ) = 0 ( r + 2 ) ( r + 3 ) = 0 r = − 2 , − 3 That is, the roots are r 1 = − 2 and r 2 = − 3 . Thus, the general solution to the homogenous equation is, c 1 e − 2 x + c 2 e − 3 x . Combine this general solution with the particular solution y p ( x ) = e x + x 2 of the non-homogenous equation to obtain y ( x ) = e x + x 2 + c 1 e − 2 x + c 2 e − 3 x . Therefore, the general solution to the differential equation y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x is, y ( x ) = e x + x 2 + c 1 e − 2 x + c 2 e − 3 x _ .

The general solution to the differential equation y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x . The general solution to the differential equation y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x is, y ( x ) = e x + x 2 + c 1 e − 2 x + c 2 e − 3 x _ . Given that, the particular solution to y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x is y p ( x ) = e x + x 2 . The associated auxiliary equation is, r 2 + 5 r + 6 r = 0 . Simplify r 2 + 5 r + 6 r = 0 as, r 2 + 2 r + 3 r + 6 r = 0 r ( r + 2 ) + 3 ( r + 2 ) = 0 ( r + 2 ) ( r + 3 ) = 0 r = − 2 , − 3 That is, the roots are r 1 = − 2 and r 2 = − 3 . Thus, the general solution to the homogenous equation is, c 1 e − 2 x + c 2 e − 3 x . Combine this general solution with the particular solution y p ( x ) = e x + x 2 of the non-homogenous equation to obtain y ( x ) = e x + x 2 + c 1 e − 2 x + c 2 e − 3 x . Therefore, the general solution to the differential equation y ″ + 5 y ′ + 6 y = 6 x 2 + 10 x + 2 + 12 e x is, y ( x ) = e x + x 2 + c 1 e − 2 x + c 2 e − 3 x _ .

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Fundamentals of Differential Equations (9th Edition)

Fundamentals of Differential Equations (9th Edition)

9th Edition

ISBN: 9780321977069

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Fundamentals of Differential Equations (9th Edition)

Ch. 4.1 - Verify that for b = 0 and Fext(t) = 0, equation...Ch. 4.1 - If Fext(t) = 0, equation (3) becomes my + by + ky...Ch. 4.1 - Show that if Fext(t) = 0, m = 1, k = 9, and b = 6,...Ch. 4.1 - Verify that y = sin 3t + 2 cos 3t is a solution to...Ch. 4.1 - Prob. 5E Ch. 4.1 - An external force F(t) = 2 cos 2t is applied to a...Ch. 4.1 - Prob. 7E Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Prob. 10E

Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 3E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - Prob. 10E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 15E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 17E Ch. 4.2 - Prob. 18E Ch. 4.2 - Prob. 19E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 21E Ch. 4.2 - Prob. 22E Ch. 4.2 - Prob. 23E Ch. 4.2 - Prob. 24E Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Prob. 33E Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Using the definition in Problem 35, prove that if...Ch. 4.2 - Prob. 37E Ch. 4.2 - Prob. 38E Ch. 4.2 - Prob. 39E Ch. 4.2 - Prob. 40E Ch. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 43E Ch. 4.2 - Solve the initial value problem: y 2y y + 2y =...Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - Prob. 5E Ch. 4.3 - Prob. 6E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 920, find a general solution. 9. y 8y...Ch. 4.3 - In Problems 920, find a general solution. 10. y +...Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - In Problems 920, find a general solution. 13. y ...Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.3 - Prob. 16E Ch. 4.3 - In Problems 920, find a general solution. 17. y y...Ch. 4.3 - In Problems 920, find a general solution. 18. 2y +...Ch. 4.3 - Prob. 19E Ch. 4.3 - In Problems 920, find a general solution. 20. y y...Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 25E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 27E Ch. 4.3 - Prob. 28E Ch. 4.3 - Prob. 29E Ch. 4.3 - Prob. 30E Ch. 4.3 - Prob. 31E Ch. 4.3 - Prob. 32E Ch. 4.3 - Prob. 33E Ch. 4.3 - Prob. 34E Ch. 4.3 - Swinging Door. 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