Given: +2+ y = et, y(0) = 1 and (a) y(1) (b) (1) Use a step size of h = 0.5 (0) = 2, Find by Heun's Method:
Given: +2+ y = et, y(0) = 1 and (a) y(1) (b) (1) Use a step size of h = 0.5 (0) = 2, Find by Heun's Method:
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:d²y
dt²
7. Given: +2+y=e¹¹, y(0) = 1 and
(a) y(1)
(b) d (1)
dt
Use a step size of h = 0.5
(0) = 2, Find by Heun's Method:
dt
Expert Solution
Step 1
Given,
(a)Let y'=z
y′′=e^(-t)-y-2z,
y(0)=1,y′(0)=2,h=0.5,y(1)=?
put dy/dt=z and differentiate w.r.t. t, we obtain d2y/dt2=dz/dt
We have system of equations
dy/dt=z=f(t,y,z)
dz/dt=exp(-t)-y-2z=g(t,y,z)
Heun method for second order differential equation
y1=y0+hf(t0,y0,z0)=1+(0.5)⋅f(0,1,2)=1+(0.5)⋅(2)=1+(1)=2
z1=z0+hg(t0,y0,z0)=2+(0.5)⋅g(0,1,2)=2+(0.5)⋅(-4)=2+(-2)=0
y2=y1+hf(t1,y1,z1)=2+(0.5)⋅f(0.5,2,0)=2+(0.5)⋅(0)=2+(0)=2
∴y(1)=2
Step by step
Solved in 2 steps
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,
