Consider the IVP dealing with nonhomogeneous second order linear differential equation with variable coefficients (x-1)y"(x)-xy'(x) + y(x) = (x-1)² e*, y(0) = 0, y'(0)=0 The functions y₁(x)=x and y₂(x)= e* 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(x) of the nonhomogeneous equation in the form y(x)=y₁ (x)v, (x) + y₂(x)₂(x), the functions v, and v₂ satisfy the system of equations OA. xv₁ (x) + e*v₂(x)=0 and v₁ (x) + e*v₂(x)=(x-1) e* OB. xv₁ (x) + e*v₂(x)=(x-1) ex and v₁ (x) + e*v₂(x) = 0 OC. xv₁ (x) + e*v₂(x) = 0 and v₁'(x) + e*v₂'(x) = (x - 1)² ex OD. xv₁'(x) + e*v₂(x) = 0 and v₁'(x) + e*v₂'(x) = (x - 1) ex OE. None of the answers given is correct
Consider the IVP dealing with nonhomogeneous second order linear differential equation with variable coefficients (x-1)y"(x)-xy'(x) + y(x) = (x-1)² e*, y(0) = 0, y'(0)=0 The functions y₁(x)=x and y₂(x)= e* 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(x) of the nonhomogeneous equation in the form y(x)=y₁ (x)v, (x) + y₂(x)₂(x), the functions v, and v₂ satisfy the system of equations OA. xv₁ (x) + e*v₂(x)=0 and v₁ (x) + e*v₂(x)=(x-1) e* OB. xv₁ (x) + e*v₂(x)=(x-1) ex and v₁ (x) + e*v₂(x) = 0 OC. xv₁ (x) + e*v₂(x) = 0 and v₁'(x) + e*v₂'(x) = (x - 1)² ex OD. xv₁'(x) + e*v₂(x) = 0 and v₁'(x) + e*v₂'(x) = (x - 1) ex OE. None of the answers given is correct
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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![Consider the IVP dealing with a nonhomogeneous second-order linear differential equation with variable coefficients.
\[
(x - 1)y''(x) - xy'(x) + y(x) = (x - 1)^2 e^x, \quad y(0) = 0, \quad y'(0) = 0
\]
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:
- **A.** \( xy_1(x) + e^xv_2(x) = 0 \) and \( v_1(x) + e^xv_2(x) = (x - 1)e^x \)
- **B.** \( xy_1(x) + e^xv_2(x) = (x - 1)e^x \) and \( v_1(x) + e^xv_2(x) = 0 \)
- **C.** \( xy_1(x) + e^xv_2(x) = 0 \) and \( v_1(x) + e^xv_2(x) = (x - 1)^2 e^x \)
- **D.** \( xy_1(x) + e^xv_2(x) = 0 \) and \( y_1(x) + e^xv_2(x) = (x - 1)e^x \)
- **E.** None of the answers given is correct](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa827acfe-a0bc-46c0-ab61-62657df3b5db%2Fe400197f-21ac-426f-ad6b-ec0334624cde%2Flxw56c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the IVP dealing with a nonhomogeneous second-order linear differential equation with variable coefficients.
\[
(x - 1)y''(x) - xy'(x) + y(x) = (x - 1)^2 e^x, \quad y(0) = 0, \quad y'(0) = 0
\]
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:
- **A.** \( xy_1(x) + e^xv_2(x) = 0 \) and \( v_1(x) + e^xv_2(x) = (x - 1)e^x \)
- **B.** \( xy_1(x) + e^xv_2(x) = (x - 1)e^x \) and \( v_1(x) + e^xv_2(x) = 0 \)
- **C.** \( xy_1(x) + e^xv_2(x) = 0 \) and \( v_1(x) + e^xv_2(x) = (x - 1)^2 e^x \)
- **D.** \( xy_1(x) + e^xv_2(x) = 0 \) and \( y_1(x) + e^xv_2(x) = (x - 1)e^x \)
- **E.** None of the answers given is correct
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