To determine The solution of the initial value problem ( x 2 D 2 + 15 x D + 49 I ) y = 0 , y ( 1 ) = 2 , y ′ ( 1 ) = − 11 and graph it. Explanation Result used: The general solution of Euler-Cauchy equation: “Suppose that, the linear ordinary differential equation of the form x 2 y ″ + a x y ′ + b y = 0 . In order to obtain the auxiliary equation, substitute y = x m , y ′ = m x m − 1 and y ″ = m ( m − 1 ) x m − 2 . The auxiliary equation is m 2 + ( a − 1 ) m + b = 0 . Then, then the general solution of the equation is of the form: Case 1: If the roots are real distinct m 1 and m 2 , then the general solution is, y h ( x ) = c 1 x m 1 + c 2 x m 2 . Case 2: If the roots are real double roots m 1 and m 1 , then the general solution is, y h ( x ) = ( c 1 + c 2 ln x ) x m 1 . Case 3: If the roots are complex a ± i b , then the general solution is, y h ( x ) = x a ( A cos ( b ln x ) + B sin ( b ln x ) ) . Calculation: Given that, the non-homogeneous linear ordinary differential equation is ( x 2 D 2 + 15 x D + 49 I ) y = 0 . It can be written as, x 2 y ″ + 15 x y ′ + 49 y = 0 . First compute the general solution y h of the non-homogeneous linear ordinary differential equation as follows. Here, the non-homogeneous linear ordinary differential equation is in form of x 2 y ″ + a x y ′ + b y = 0 . The auxiliary equation of x 2 y ″ + 15 x y ′ + 49 y = 0 is m 2 + ( 15 − 1 ) m + 49 = 0 . m 2 + ( 15 − 1 ) m + 49 = 0 m 2 + 14 m + 49 = 0 ( m + 7 ) 2 = 1 m = − 7 , − 7 Thus, the roots of the auxiliary equation m 2 + ( 1 − 1 ) m − 9 = 0 are − 7 , − 7 . Here, the roots are real double roots. So, the general solution y h of the non-homogeneous linear ordinary differential equation is, y h ( x ) = ( A + B ln x ) x − 7 . The solution of the given ordinary differential equation is computed as follows. y ( x ) = y h ( x ) + y p ( x ) = ( A + B ln x ) x − 7 In order obtain the value of A and B , use initial condition y ( 1 ) = 2 , y ′ ( 1 ) = − 11 as follows. y ( x ) = ( A + B ln x ) x − 7 y ( 1 ) = ( A + B ln 1 ) ( 1 ) − 7 2 = ( A + B ( 0 ) ) A = 2 Differentiate y ( x ) = ( A + B ln x ) x − 7 with respect to x as shown below

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2, Problem 22RQ

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The solution of the initial value problem (x2D2+15xD+49I)y=0,y(1)=2,y′(1)=−11 and graph it.

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Chapter 2, Problem 21RQ

Chapter 2, Problem 23RQ

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