Based on the above, find the general solution of the following differential equation y""-y" - 2y' = (100x - 25) cos2x, y = y(x).
Based on the above, find the general solution of the following differential equation y""-y" - 2y' = (100x - 25) cos2x, y = y(x).
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
![EXERCISE 5
The method of determinable coefficients for finding a solution of a non-homogeneous linear ordinary differential
equation with constant coefficients can be applied when the non-homogeneous term f(x) of the equation has the form:
f(x) = ex [Pm(x)cos(bx)+Q:(x)sin(bx)],
where a, beR and Pm(x), Q(x) polynomials of x degree m and r respectively. In this case the o.d.e. that we want to
solve is amenable to a solution of the form:
yu(x)=x³ex [R₂(x)cos(bx)+T:(x)sin(bx)],
where s=max{m,r}, R.(x),T:(x) are complete polynomials of x degrees whose coefficients are to be determined and
is the multiplicity with which the number a+bi appears as a root of the characteristic equation of the corresponding
homogeneous o.d.e.
(Consequently, if a+bi is not a root of the corresponding characteristic equation, then = 0, if it is a simple root then
π = 1, if it is a double root then -2, and so on).
In the more special case where the non-homogeneous o.d.e. has the form
y() (x)+ay(n-1)(x)+...+an-ky)(x) = f(x) (1)
where, an-k+1=..=an-1-an=0 and f(x)=20x² +21x²-¹+...+2μu-Ix+u, then to find a partial solution of (1) we set
yk)(x) = box+bx-1+...+bu-ix+bu and calculate the coefficients bi, ie{0,1,...,}.
Then with k-successive integrations we calculate the partial solution of (1).
Based on the above, find the general solution of the following differential equation
y"-y" - 2y' = (100x - 25) cos2x,
y = y(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4d74cef-f64d-4ed0-a004-41e9fb8cce2f%2F0496ad3d-ed33-442f-8020-7ff9479ea069%2F0oe4k7h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:EXERCISE 5
The method of determinable coefficients for finding a solution of a non-homogeneous linear ordinary differential
equation with constant coefficients can be applied when the non-homogeneous term f(x) of the equation has the form:
f(x) = ex [Pm(x)cos(bx)+Q:(x)sin(bx)],
where a, beR and Pm(x), Q(x) polynomials of x degree m and r respectively. In this case the o.d.e. that we want to
solve is amenable to a solution of the form:
yu(x)=x³ex [R₂(x)cos(bx)+T:(x)sin(bx)],
where s=max{m,r}, R.(x),T:(x) are complete polynomials of x degrees whose coefficients are to be determined and
is the multiplicity with which the number a+bi appears as a root of the characteristic equation of the corresponding
homogeneous o.d.e.
(Consequently, if a+bi is not a root of the corresponding characteristic equation, then = 0, if it is a simple root then
π = 1, if it is a double root then -2, and so on).
In the more special case where the non-homogeneous o.d.e. has the form
y() (x)+ay(n-1)(x)+...+an-ky)(x) = f(x) (1)
where, an-k+1=..=an-1-an=0 and f(x)=20x² +21x²-¹+...+2μu-Ix+u, then to find a partial solution of (1) we set
yk)(x) = box+bx-1+...+bu-ix+bu and calculate the coefficients bi, ie{0,1,...,}.
Then with k-successive integrations we calculate the partial solution of (1).
Based on the above, find the general solution of the following differential equation
y"-y" - 2y' = (100x - 25) cos2x,
y = y(x).
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 3 steps with 2 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,
