Math 128C: Practice Midterm
Name:
Question Points Score
1
2
3
Total
Start Time: 11:00 am
End Time: 11:50 am
Instructions: You have 50 minutes to complete this exam. No calculators or other electronic devices
are permitted. Justify your answers in order to receive full credit.
1. (a) (5 points) Let u(x) be a function, and suppose h > 0. Determine the constants a, b, and c such
that the following expression is a second order approximation to u′(x):
au(x− h) + bu(x) + cu(x+ 3h)
h
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(b) (5 points) Consider the third order differential equation
y′′′ + 4y′′ + 3y′ + 6y = et
with initial conditions
y(0) = 1, y′(0) = 0, y′′(0) = 1.
Convert this equation an equivalent system of first order differential equations, and specify the
initial conditions.
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2. Consider the following multistep method for solving
dy
dt
= f(t, y):
yn+1 = −yn + 2yn−1 + h
(
5
2
f(tn, yn) +
1
2
f(tn−1, yn−1)
)
.
(a) (4 points) Is this method consistent?
(b) (4 points) Is this method zero-stable?
(c) (2 points) Is this method convergent?
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3. Consider the following three stage method for solving the initial value problem dydt = f(t, y) with
initial condition y(0) = α:
k1 = f(tn, yn)
k2 = f(tn + h, yn + hk1)
k3 = f(tn + h, yn + hk2)
yn+1 = yn +
h
4
(2k1 + k2 + k3)
(a) (5 points) If f(t, y) = λy, write down a formula for yn+1 in terms of yn, h, and λ.
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(b) (5 points) Determine the condition in terms of h and λ that define the region of absolute
stability.
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