Tutorial 8: Numerical Methods
Writing higher-order ODEs as a first-order system
1. Write the following ODEs as first-order systems.
(a) θ ′′ +
g
L
sin θ = 0 (Pendulum).
(b) y ′′ − µ(1 − y 2 )y ′ + y = 0, µ > 0 constant (Van der Pol oscillator).
2
d y
dy
2
2
(c) x2 dx
2 + x dx + (x − ν )y = 0, ν ∈ R (Bessel equation).
Euler’s Method To approximate the solution to y ′ = f (t, y), y(t0 ) = y0 , Euler’s method approximates the solution y(t) by y(tn ) with tn = t0 + nh and
y0 = y(t0 ),
yn+1 = yn + hf (tn , yn ).
2. Set up the recursion formulas for Euler’s method for each system found above with the
specified initial condition.
(a) θ ′′ +
g
L
sin θ = 0, θ(0) = 0, θ ′ (0) = 1.
(b) y ′′ − µ(1 − y 2 )y ′ + y = 0, y(0) = 2, y ′ (0) = 1.
1
2
dy
d y
2
2
′
(c) x2 dx
2 + x dx + (x − ν )y = 0, y(0) = 1, y (0) = 0.
3. The first iterate of Euler’s method for y ′ = y, y(0) = 1 and x′ = x(1 − x), x(0) = 0.8 are
plotted below with h = 0.5. Complete the sketches (no calculations needed).
8
1
Euler
Exact
4
x
y
6
0.9
Euler
Exact
2
0
0.8
0
0.5
1
t
1.5
2
0
0.5
1
t
1.5
2
(a) What can you conclude about the error in Euler’s method when the solution is concave
up or concave down? Do you over- or under-estimate the solution?
(b) How does the Improved Euler method address this issue?
Computer time
1. Use Matlab (or similar) to implement Euler’s method for y ′ = −10y, y(0) = 1. What happens
for different choices of the time step h?
2. Use Matlab (or similar) to implement Euler’s method for y ′′ − µ(1 − y 2 )y ′ + y = 0, y(0) = 2,
y ′ (0) = 1. What happens for different choices of the time step h? What happens for different
choices of µ? Try µ = 1 and µ = 100.
3. If you’re annoyed with how unstable Euler’s method is, try the Backward Euler Method. For
an ODE y ′ = f (t, y), y(t0 ) = y0 , the Backward Euler Method computes approximations using
y0 = y(t0 ),
yn+1 = yn + hf (tn+1 , yn+1 ).
That is, during each time step, the (possibly nonlinear) equation
yn+1 − hf (tn+1 , yn+1 ) = yn
must be solved for yn+1 . Do you know a method that you can use to solve nonlinear equations?
2