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Determine the solution for the following higher-order nonhomogeneous differential equation. Also, determine the constant coefficients if the initial condition is given. 1) y" - 2y' - 3y = 3x² + 4x - 5 if y(0) = 9 y'(0)=-4

Determine the solution for the following higher-order nonhomogeneous differential equation. Also, determine the constant coefficients if the initial condition is given. 1) y" - 2y' - 3y = 3x² + 4x - 5 if y(0) = 9 y'(0)=-4

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Determine the solution for the following higher-order nonhomogeneous differential equation. Also, determine the constant coefficients if the initial condition is given. 1) y" - 2y' - 3y = 3x² + 4x - 5 if y(0) = 9 y'(0)=-4
Transcribed Image Text:Determine the solution for the following higher-order nonhomogeneous differential equation. Also, determine the constant coefficients if the initial condition is given. 1) y" - 2y' - 3y = 3x² + 4x - 5 if y(0) = 9 y'(0)=-4
Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form 1 ay" +by' + cy= G(x) where a, b, and c are constants and G is a continuous function. The related homogeneous equation 2 ay" + by' + cy=0 is called the complementary equation and plays an important role in the solution of the original nonhomogeneous equation [. The Method of Undetermined Coefficients We first illustrate the method of undetermined coefficients for the equation ay" + by' + cy= G(x) where G(x) is a polynomial. It is reasonable to guess that there is a particular solution y, that is a polynomial of the same degree as G because if y is a polynomial, then ay" + by' + cy is also a polynomial. We therefore substitute y(x) = a polynomial (of the same degr as G) into the differential equation and determine the coefficients.
Transcribed Image Text:Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form 1 ay" +by' + cy= G(x) where a, b, and c are constants and G is a continuous function. The related homogeneous equation 2 ay" + by' + cy=0 is called the complementary equation and plays an important role in the solution of the original nonhomogeneous equation [. The Method of Undetermined Coefficients We first illustrate the method of undetermined coefficients for the equation ay" + by' + cy= G(x) where G(x) is a polynomial. It is reasonable to guess that there is a particular solution y, that is a polynomial of the same degree as G because if y is a polynomial, then ay" + by' + cy is also a polynomial. We therefore substitute y(x) = a polynomial (of the same degr as G) into the differential equation and determine the coefficients.
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