To determine The general solution of the nonhomogeneous linear ordinary differential equation y ″ + 3 y ′ + 2 y = 12 x 2 . Explanation Formula used: Method of undetermined coefficients: Consider the linear ordinary differential equation y ″ + a y ′ + b y = r ( x ) . 1. If the term in r ( x ) is of the form k e γ x , then the choice for y p ( x ) is C e γ x . 2. If the term in r ( x ) is of the form k x n ( n = 0 , 1 , … ) , then the choice for y p ( x ) is K n x n + K n − 1 x n − 1 + ⋯ + K 1 x + K 0 . 3. If the term in r ( x ) is of the form k cos ω x or k sin ω x , then the choice for y p ( x ) is K cos ω x + M sin ω x . 4. If the term in r ( x ) is of the form k e α x cos ω x or k e α x sin ω x , then the choice for y p ( x ) is e α x ( K cos ω x + M sin ω x ) . Result used: Basic rule : In this rule, determine the undetermined coefficients of y p by substituting y p and its derivatives in y ″ + a y ′ + b y = r ( x ) . Calculation: The given ordinary differential equation is y ″ + 3 y ′ + 2 y = 12 x 2 . Obtain the solution y h of the homogeneous differential equation y ″ + 3 y ′ + 2 y = 0 as follows. The auxiliary equation of y ″ + 3 y ′ + 2 y = 0 is m 2 + 3 m + 2 = 0 . Now obtain the roots of the auxiliary equation m 2 + 3 m + 2 = 0 as shown below. m 2 + 3 m + 2 = 0 m 2 + m + 2 m + 2 = 0 m ( m + 1 ) + 2 ( m + 1 ) = 0 ( m + 1 ) ( m + 2 ) = 0 m = − 1 , − 2 Here the roots are real and distinct. Therefore the solution y h is y h = c 1 e − x + c 2 e − 2 x . Now obtain the particular solution y p as follows. Here the term r ( x ) is 12 x 2 which is of the form k x n , and hence the choice for y p becomes K 2 x 2 + K 1 x + K 0 . That is, y p = K 2 x 2 + K 1 x + K 0 . Differentiate y p = K 2 x 2 + K 1 x + K 0 with respect to x and obtain the first derivative as shown below. y ′ p = d d x ( K 2 x 2 + K 1 x + K 0 ) = 2 K 2 x + K 1 + 0 = 2 K 2 x + K 1 Again differentiate y ′ p and obtain the second derivative as follows. y ″ p = d d x ( 2 K 2 x + K 1 ) = 2 K 2 + 0 = 2 K 2 By basic rule, substitute the obtained derivatives in y ″ + 3 y ′ + 2 y = 0 and determine the undetermined coefficients as shown below

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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Chapter 2.7, Problem 3P

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The general solution of the nonhomogeneous linear ordinary differential equation y″+3y′+2y=12x2.

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Chapter 2.7, Problem 2P

Chapter 2.7, Problem 4P

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