Math 151B Homework #4 – not turned in
1. Adams-Bashforth two point method.
Solve the initial value problem
y0=−ty, 0≤t≤4, y(0) = 1
using the Adams-Bashforth two point method.
Do the same with the midpoint method.
Both of these are 2nd order methods. Which one requires more function evaluations?
2. Second order equation
The motion of a pendulum of length `is described by the equation
θ00 =−g
`sin θ
where θis the angular deviation from vertical.
Let g/` = 1, so we have θ00 =−sin θ.
a. Linearize this equation (i.e. approximate sinθwith a linear function) and solve exactly.
b. Assume the pendulum starts at an angle of 15◦with no initial velocity. Solve the non-
linear pendulum equation numerically using Euler’s method and compare to the linearized
solution.
c. Is explicit Euler’s method a good method to use in this case. Why? (Hint: approximate
the region of stability)
d. Challenge! Write an implicit Euler’s method to solve this system.