Explanation Procedure used: Consider the differential equation a t 2 y ″ ( t ) + b t y ′ ( t ) + c y = 0 . The associated auxiliary equation is a r 2 + ( b − a ) r + c = 0 . Solving the auxiliary equation gives the roots r 1 and r 2 . (a) If the auxiliary equation has distinct real roots r 1 and r 2 , then y ( t ) = c 1 t r 1 + c 2 t r 2 is a general solution. (b) If the auxiliary equation has repeated root r , then y ( t ) = c 1 t r + c 2 t r ln t is a general solution. (b) If the auxiliary equation has complex root r = α ± i β , then y ( t ) = c 1 t α cos ( β ln t ) + c 2 t α sin ( β ln t ) is a general solution. Calculation: The given differential equation is, t 2 y ″ ( t ) + 7 t y ′ ( t ) − 7 y ( t ) = 0 . Comparing the differential equation t 2 y ″ ( t ) + 7 t y ′ ( t ) − 7 y ( t ) = 0 with the general form a t 2 y ″ ( t ) + b t y ′ ( t ) + c y = 0 gives a = 1 , b = 7 and c = − 7

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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Chapter 4.7, Problem 9E

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The general solution to the Cauchy-Euler equation t2y″(t)+7ty′(t)−7y(t)=0 for t>0.

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Chapter 4.7, Problem 8E

Chapter 4.7, Problem 10E

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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

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The general solution to the Cauchy-Euler equation t 2 y ″ ( t ) + 7 t y ′ ( t ) − 7 y ( t ) = 0 for t &gt; 0 .

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The general solution to the Cauchy-Euler equation t2y″(t)+7ty′(t)−7y(t)=0 for t>0.

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The general solution to the Cauchy-Euler equation t 2 y ″ ( t ) + 7 t y ′ ( t ) − 7 y ( t ) = 0 for t > 0 .