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
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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