Explanation Given: The second order differential equation y ″ − ( 2 α − 1 ) y ′ + α ( α − 1 ) y = 0 . Formula used: The auxiliary equation of a linear homogeneous differential equation of the form a 0 y ″ + a 1 y ′ + a 2 y = 0 with constant coefficients is given by a 0 r 2 + a 1 r + a 2 = 0 . The general solution of the auxiliary equation with complex roots is given by y = e α x ( c 1 sin β x + c 2 cos β x ) . The general solution of the auxiliary equation with double roots r is given by y = e r x ( c 1 + c 2 x ) . The general solution of the auxiliary equation with two distinct roots r 1 and r 2 , is given by y = c 1 e r 1 x + c 2 e r 2 x . Calculation: The given differential equation is y ″ − ( 2 α − 1 ) y ′ + α ( α − 1 ) y = 0 . Compare the given equation with a 0 y ″ + a 1 y ′ + a 2 y = 0 and obtain the values of a 0 , a 1 , a 2 as shown below: a 0 = 1 , a 1 = − ( 2 α − 1 ) , a 2 = α ( α − 1 ) . Write the auxiliary equation of the homogeneous equation y ″ − ( 2 α − 1 ) y ′ + α ( α − 1 ) y = 0 as, r 2 − ( 2 α − 1 ) r + α ( α − 1 ) = 0 Solve the auxiliary equation and obtain the roots as follows: r 2 − ( 2 α − 1 ) r + α ( α − 1 ) = 0 r = α , α − 1 Thus, the auxiliary equation has three distinct roots r = α , α − 1

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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The solution of the initial value problem and determine the value of α as the solution approaches zero as t→∞ and when solution becomes unbounded.

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The solution of the initial value problem and determine the value of α as the solution approaches zero as t → ∞ and when solution becomes unbounded.

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The solution of the initial value problem and determine the value of α as the solution approaches zero as t→∞ and when solution becomes unbounded.

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