Explanation Given: Given differential equation is t 2 x ″ ( t ) − 12 x ( t ) = 0 ; x ( 1 ) = 3 , x ′ ( 1 ) = 5 Approach: A second-order homogeneous equation in the form a x 2 y ″ ( x ) + b x y ′ ( x ) + c y ( x ) = 0 is called Cauchy - Euler equation, where a ≠ 0 and b and c are real numbers. If the equation has a singular point then it has solution in the form of y = x r . If the roots of equation a r 2 + b r + c = 0 are real such that r 1 ≠ r 2 then general solution is y ( t ) = c 1 e r 1 t + c 2 e r 2 t Calculation: Consider the problem. t 2 x ″ ( t ) − 12 x ( t ) = 0 ... ( 1 ) Let, w ( r , t ) = t r and L is the left hand side of the equation. Differentiate w ( r , t ) = t r with respect to r upto second derivative. d 2 d r 2 { L [ w ] ( t ) } = d 2 d r 2 ( t ) r = d d r r ( t ) t r − 1 = r ( r − 1 ) t r − 2 Substitute r ( r − 1 ) x r − 2 for x ″ , r ( x ) x r − 1 for x ′ , and ( t ) r for x in equation ( 1 ) . L [ w ] ( t ) = [ t 2 r ( r − 1 ) t r − 2 − 12 t r ] = [ r ( r − 1 ) t r − 12 t r ] = [ r ( r − 1 ) − 12 ] ( t ) r = [ r 2 − r − 12 ] ( x ) r Equate the coefficient of ( x ) r with zero. r 2 − r − 12 = 0 ( 2 ) Simplify equation ( 2 )

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

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

Publisher: PEARSON

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Solution Summary: The author explains the Cauchy - Euler equation, where ane 0, b and c are real numbers.

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