thon of parameters works for higher-order nonhomogeneous linear differential equations too. problem shows you how to solve y" + P(1)y" + QU)y + R()y = g(1). t y,(), y2(), and y,(1) be three linearly independent solutions to the homogeneous quation y" + P(t)y" + Q(M)y + R¢)y = 0. hen assume that a particular solution is of the form (1) = 4,(1)y,(1) + u;(1)y2(1) + us(1)ys(t). ow solve the following system of linear equations for ư,(1), u,(1), and u'(1): ( u,yn + u,yz + u;ys == 0 lu,x" + u,y% + u,y% = g(1) he first two equations are clever assumptions, and the last equation comes from plugging = u1 yı + usy2 + u;3% into the nonhomogeneous equation.) nce you have , (1), ư,(1), and ư',(1) then integrate to find u, (1), uz(1), and u,(1). Then STION: e this order-three version of variation of parameters to find the general solution: y" +2y" – y – 2y = e'
thon of parameters works for higher-order nonhomogeneous linear differential equations too. problem shows you how to solve y" + P(1)y" + QU)y + R()y = g(1). t y,(), y2(), and y,(1) be three linearly independent solutions to the homogeneous quation y" + P(t)y" + Q(M)y + R¢)y = 0. hen assume that a particular solution is of the form (1) = 4,(1)y,(1) + u;(1)y2(1) + us(1)ys(t). ow solve the following system of linear equations for ư,(1), u,(1), and u'(1): ( u,yn + u,yz + u;ys == 0 lu,x" + u,y% + u,y% = g(1) he first two equations are clever assumptions, and the last equation comes from plugging = u1 yı + usy2 + u;3% into the nonhomogeneous equation.) nce you have , (1), ư,(1), and ư',(1) then integrate to find u, (1), uz(1), and u,(1). Then STION: e this order-three version of variation of parameters to find the general solution: y" +2y" – y – 2y = e'
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:INTRODUCTION:
Variation of parameters works for higher-order nonhomogeneous linear differential equations too.
This problem shows you how to solve y" + P(0)y" + QU)y + R0)y = g(1).
• Let y, (1), y2(1), and y, (1) be three linearly independent solutions to the homogeneous
equation y" + P(t)y" + Q(t)y' + R(t)y = 0.
• Then assume that a particular solution is of the form
y(1) = u,(1)y;(1) + tuz(1)y½(1) + u3(1)y,(1).
• Now solve the following system of linear equations for u (1), u,(1), and u (1):
4 + u½ + u¥% = 0
uy" + u,y% + u,y% = g(1)
(The first two equations are clever assumptions, and the last equation comes from plugging
y = u, yı + u,y + u;y3 into the nonhomogeneous equation.)
• Once you have ư (t), u,(1), and ư (1) then integrate to find u, (t), uz(f), and u;(1). Then
y = u,yı + u2y2 + u;y3.
QUESTION:
Use this order-three version of variation of parameters to find the general solution:
y" + 2y" – y – 2y = e'
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
