Explanation Approach: A linear second-order equation that can be expressed in the form a t 2 y ″ ( t ) + b t y ′ ( t ) + c y = f ( t ) where a , b , and c are constants, is called a Cauchy-Euler equation. The associated characteristic equation of Cauchy-Euler equation is given by, a r 2 + ( b − a ) r + c = 0 . The roots of this associated characteristic equation gives the general solution of the differential equation. Calculation: The given Cauchy-Euler equation is, t 2 y ″ ( t ) − 3 t y ′ ( t ) + 4 y ( t ) = 0... ( 1 ) Standard Cauchy-Euler equation is, a t 2 y ″ ( t ) + b t y ′ ( t ) + c y ( t ) = f ( t ) Then the characteristic equation for the Cauchy-Euler equation is, a r 2 + ( b − a ) r + c = 0... ( 2 ) Compare equation ( 1 ) with standard Cauchy-Euler equation this gives a = 1 , b = − 3 , and c = 4 . Substitute 1 for a , − 3 for b and 4 for c in equation ( 2 ) .

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ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

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Chapter 4.7, Problem 14E

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A general solution to the given Cauchy-Euler equation t2y″(t)−3ty′(t)+4y(t)=0 for t>0.

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Chapter 4.7, Problem 13E

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A general solution to the given Cauchy-Euler equation t 2 y ″ ( t ) − 3 t y ′ ( t ) + 4 y ( t ) = 0 for t > 0 .

Solution Summary: The author explains the Cauchy-Euler equation and the associated characteristic equation, which gives the general solution of the differential equation.

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To find:

A general solution to the given Cauchy-Euler equation t2y″(t)−3ty′(t)+4y(t)=0 for t>0.

Want to see the full answer?

Chapter 4 Solutions

A general solution to the given Cauchy-Euler equation t 2 y ″ ( t ) − 3 t y ′ ( t ) + 4 y ( t ) = 0 for t &gt; 0 .

Solution Summary: The author explains the Cauchy-Euler equation and the associated characteristic equation, which gives the general solution of the differential equation.

Concept explainers

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To find: A general solution to the given Cauchy-Euler equation t2y″(t)−3ty′(t)+4y(t)=0 for t>0.

Want to see the full answer?

Chapter 4 Solutions

A general solution to the given Cauchy-Euler equation t 2 y ″ ( t ) − 3 t y ′ ( t ) + 4 y ( t ) = 0 for t > 0 .

To find:

A general solution to the given Cauchy-Euler equation t2y″(t)−3ty′(t)+4y(t)=0 for t>0.