Name:
SECOND PRACTICE TEST FOR MIDTERM 1, MATH 216
Problem 1.
Answer the following questions.
a) 5 pts. Consider
(
tanx
)
y
0
+
1
x
+ 2
y
=
x
3
Fine the largest interval of
x
containing
x
=

1 where a unique solution of this initialvalue
problem is guaranteed to exist based on what we have learned in the course. Explain your answer.
Do not solve the diﬀerential equation.
b) 5 pts Consider the autonomous system
dx/dt
= 7
x
(5

x
) Draw a phase diagram and determine
whether
x
= 0 is a stable or unstable equilibrium.
c) 5 pts Consider the initial value problem
dy/dx
=

1
/
(1 +
x
2
)
, y
(0) = 1
Suppose we apply the improved Euler method to approximate the solution on the interval [0
,
2]
with step size
h
= 0
.
02. Comparing the result of this numerical calculation with the exact solution
at
x
= 2 suppose that the cumulative error at
x
= 2 is 0.2. From this information, predict a step size
which might be reasonably expected to result in a cumulative error 1/4 that obtained for
h
= 0
.
02
.
d) 5 pts Consider
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 Spring '07
 Stenstones?
 Math, pts, Boundary value problem

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