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We want to find functions u₁(x) and u₂(x) such that yp = U₁Y₁+U₂Y₂ is a particular solution. We can find the derivatives of these functions as follows. W U₁ = = -e-2x(2 + ex)-1 -e-3x W₂ e-x(2 + ex)-1 -e-3x

We want to find functions u₁(x) and u₂(x) such that yp = U₁Y₁+U₂Y₂ is a particular solution. We can find the derivatives of these functions as follows. W U₁ = = -e-2x(2 + ex)-1 -e-3x W₂ e-x(2 + ex)-1 -e-3x

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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We have found the following Wronskians. W = -e-³x; W₁ = - We want to find functions u₁(x) and u₂(x) such that yp = U₁Y₁ + U₂y₂ is a particular solution. We can find the derivatives of these functions as follows. W = 1 W -e-2x(2 + ex)-1 -e-3x -e-²x(2 + ex)-¹; W₂ = e¯x(2 + ex)-¹ W₂ W ex(2 + ex)-1 -e-3x
Transcribed Image Text:We have found the following Wronskians. W = -e-³x; W₁ = - We want to find functions u₁(x) and u₂(x) such that yp = U₁Y₁ + U₂y₂ is a particular solution. We can find the derivatives of these functions as follows. W = 1 W -e-2x(2 + ex)-1 -e-3x -e-²x(2 + ex)-¹; W₂ = e¯x(2 + ex)-¹ W₂ W ex(2 + ex)-1 -e-3x
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We have found the following complementary function for the given differential equation. Y=₁₂x+ c₂e-2x We have also found that for y₁ = ex, y₂ = e-²x, ‚ U₁ = In(2 + e*), and U₂ = 2 In(2 + e*) – e*, a particular solution for the equation is given by y₁ = U₁Y₁+U₂Y₂ To finish, use the fact that y = y + y is the general solution of the nonhomogeneous differential equation to solve. y(x) =
Transcribed Image Text:We have found the following complementary function for the given differential equation. Y=₁₂x+ c₂e-2x We have also found that for y₁ = ex, y₂ = e-²x, ‚ U₁ = In(2 + e*), and U₂ = 2 In(2 + e*) – e*, a particular solution for the equation is given by y₁ = U₁Y₁+U₂Y₂ To finish, use the fact that y = y + y is the general solution of the nonhomogeneous differential equation to solve. y(x) =
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