Explanation The differential equation is, a y ″ + b y ′ + c y = 0 . The auxiliary equation is, a r 2 + b r + c = 0 . Let L be the differential operator. If r 0 is a repeated root, then L [ W ] ( t ) = a ( r − r 0 ) 2 e r t ... ( 1 ) Substitute r 0 for r in equation ( 1 ) to get the solution, y 1 ( t ) = e r 0 t ... ( 2 ) For the second linearly independent solution, Take partial derivate of ( 1 ) with respect to r and then substitute r 0 for r to get, ∂ ∂ r { L [ W ] ( x ) } | r = r 0 = { a ( r − r 0 ) 2 e r t + 2 a ( r − r 0 ) e h r t } | r = r 0 = 0 Now, W ( r , t ) = e r t has continuous partial derivates of all orders with respect to r and t both. So, ∂ 3 W ∂ r ∂ t 2 = ∂ 2 ω ∂ t 2 ∂ r and ∂ 3 ω ∂ r ∂ t = ∂ 2 ω ∂ t ∂ r For L ,

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

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

Publisher: Addison Wesley

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Chapter 8.5, Problem 18E

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The second linearly independent solution of the equation ay″+by′+cy=0, where a, b, and c are constants with r0 as a repeated root of the auxiliary equation ar2+br+c=0 is y2(t)=ter0t.

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Chapter 8.5, Problem 17E

Chapter 8.5, Problem 19E

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Chapter 8 Solutions

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The second linearly independent solution of the equation a y ″ + b y ′ + c y = 0 , where a , b , and c are constants with r 0 as a repeated root of the auxiliary equation a r 2 + b r + c = 0 is y 2 ( t ) = t e r 0 t .

Solution Summary: The author explains the second linearly independent solution of the equation ay′′+b

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To show:

The second linearly independent solution of the equation ay″+by′+cy=0, where a, b, and c are constants with r0 as a repeated root of the auxiliary equation ar2+br+c=0 is y2(t)=ter0t.

Want to see the full answer?

Chapter 8 Solutions

The second linearly independent solution of the equation a y ″ + b y ′ + c y = 0 , where a , b , and c are constants with r 0 as a repeated root of the auxiliary equation a r 2 + b r + c = 0 is y 2 ( t ) = t e r 0 t .

Solution Summary: The author explains the second linearly independent solution of the equation ay′′+b

Videos

To show: The second linearly independent solution of the equation ay″+by′+cy=0, where a, b, and c are constants with r0 as a repeated root of the auxiliary equation ar2+br+c=0 is y2(t)=ter0t.

Want to see the full answer?

Chapter 8 Solutions

To show:

The second linearly independent solution of the equation ay″+by′+cy=0, where a, b, and c are constants with r0 as a repeated root of the auxiliary equation ar2+br+c=0 is y2(t)=ter0t.