To determine The general solution of the equation ( x 2 D 2 − 4 x D + 6 I ) y = C . Explanation Result used: The general solution of Euler-Cauchy equation: The general solution depends on the roots of the auxiliary equation as shown below. Case 1: If the roots m 1 and m 2 are distinct real roots, then the general solution is, y = c 1 x m 1 + c 2 x m 2 . Case 2: If root m 1 is a real double root, then the general solution is, y = ( c 1 + c 2 ln x ) x m . Case 3: If the roots m 1 = a + i b and m 2 = a − i b are complex conjugate, then the general solution is, y = x a ( A cos ( b ln x ) + B sin ( b ln x ) ) . Calculation: The given equation is ( x 2 D 2 − 4 x D + 6 I ) y = C . The Euler Cauchy equation of ( x 2 D 2 − 4 x D + 6 I ) y = C is x 2 y ″ − 4 x y ′ + 6 y = 0 . First substitute y = x m , y ′ = m x m − 1 and y ″ = m ( m − 1 ) x m − 2 in x 2 y ″ − 4 x y ′ + 6 y = 0 as shown below

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ISBN: 2819770198774

Author: Kreyszig

Publisher: WILEY CONS

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Chapter 2.5, Problem 7P

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The general solution of the equation (x2D2−4xD+6I)y=C.

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Chapter 2.5, Problem 6P

Chapter 2.5, Problem 8P

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