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(b) The functions v₁' and v₂' are given by O A. v ₁ '(x) = - ex v₁'(x) = - OB. v₁'(x) = ex and v₂'(x) = 0 OC. v₁'(x) = -(x-1) e* and v₂'(x) = x(x - 1) OD. v₁'(x) = - e* and v₂'(x)=x OE. None of the answers is correct e* and v₂'(x) = 1

(b) The functions v₁' and v₂' are given by O A. v ₁ '(x) = - ex v₁'(x) = - OB. v₁'(x) = ex and v₂'(x) = 0 OC. v₁'(x) = -(x-1) e* and v₂'(x) = x(x - 1) OD. v₁'(x) = - e* and v₂'(x)=x OE. None of the answers is correct e* and v₂'(x) = 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Part B
The problem provides options for derivatives of two functions, \( v_1'(x) \) and \( v_2'(x) \). The options are as follows: **(b) The functions \( v_1' \) and \( v_2' \) are given by:** - **A.** \( v_1'(x) = -e^x \) and \( v_2'(x) = 1 \) - **B.** \( v_1'(x) = e^x \) and \( v_2'(x) = 0 \) - **C.** \( v_1'(x) = -(x-1)e^x \) and \( v_2'(x) = x(x-1) \) - **D.** \( v_1'(x) = -e^x \) and \( v_2'(x) = x \) - **E.** None of the answers is correct The task is to determine which option correctly describes the functions based on their derivatives as given in the problem context.
Transcribed Image Text:The problem provides options for derivatives of two functions, \( v_1'(x) \) and \( v_2'(x) \). The options are as follows: **(b) The functions \( v_1' \) and \( v_2' \) are given by:** - **A.** \( v_1'(x) = -e^x \) and \( v_2'(x) = 1 \) - **B.** \( v_1'(x) = e^x \) and \( v_2'(x) = 0 \) - **C.** \( v_1'(x) = -(x-1)e^x \) and \( v_2'(x) = x(x-1) \) - **D.** \( v_1'(x) = -e^x \) and \( v_2'(x) = x \) - **E.** None of the answers is correct The task is to determine which option correctly describes the functions based on their derivatives as given in the problem context.
Consider the IVP dealing with nonhomogeneous second order linear differential equations with variable coefficients \((x - 1)y''(x) - xy'(x) + y(x) = (x - 1)^2 e^x\), \[y(0) = 0, \, y'(0) = 0\] The functions \(y_1(x) = x\) and \(y_2(x) = e^x\) are independent solutions of the associated homogeneous equation \((x - 1)y''(x) - xy'(x) + y(x) = 0\). (a) When using the method of variation of parameters to find a particular solution \(y_p(x)\) of the nonhomogeneous equation in the form \(y_p(x) = y_1(x)v_1(x) + y_2(x)v_2(x)\), the functions \(v_1\) and \(v_2\) satisfy the system of equations - \( \text{A. } x v_1'(x) + e^x v_2'(x) = 0 \quad \text{and} \quad v_1'(x) + e^x v_2'(x) = (x - 1)e^x \) - \( \text{B. } e^x v_1'(x) + x v_2'(x) = (x - 1)^2 e^x \quad \text{and} \quad v_1'(x) + e^x v_2'(x) = 0 \)
Transcribed Image Text:Consider the IVP dealing with nonhomogeneous second order linear differential equations with variable coefficients \((x - 1)y''(x) - xy'(x) + y(x) = (x - 1)^2 e^x\), \[y(0) = 0, \, y'(0) = 0\] The functions \(y_1(x) = x\) and \(y_2(x) = e^x\) are independent solutions of the associated homogeneous equation \((x - 1)y''(x) - xy'(x) + y(x) = 0\). (a) When using the method of variation of parameters to find a particular solution \(y_p(x)\) of the nonhomogeneous equation in the form \(y_p(x) = y_1(x)v_1(x) + y_2(x)v_2(x)\), the functions \(v_1\) and \(v_2\) satisfy the system of equations - \( \text{A. } x v_1'(x) + e^x v_2'(x) = 0 \quad \text{and} \quad v_1'(x) + e^x v_2'(x) = (x - 1)e^x \) - \( \text{B. } e^x v_1'(x) + x v_2'(x) = (x - 1)^2 e^x \quad \text{and} \quad v_1'(x) + e^x v_2'(x) = 0 \)
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