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Math 1B
Third Exam
Name:_________________________
SID:_____________________
(last, first)
1.
The direction field for a differential equation is shown.
a) Suppose
(0)
0
y
=
.
Using Euler’s method with a stepsize of 1, what would the approximation for
(2)
y
be?
(7 points)
(0) 1.5
y
≈
b) Would this be an underestimate or an overestimate? Explain your answer.
(1 point)
This is an underestimate.
The slopes are increasing, so the function is
concave up.
c) Based on the picture above, is the differential equation autonomous?
Justify your answer.
(2 points)
Yes, the equation is autonomous.
For any fixed
y
value, the slope is
constant.
It does not depend on
x
.
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View Full Document2.
a) Give an example of a first order differential equation that is separable but not linear.
Solve your equation.
22
2
2
11
dy
x
ydy
xdx
y
x
C
y
x
C
dx
y
=→
= → = +→ − =
b) Give an example of first order differential equation that is linear, but not separable.
Solve your equation.
()
xxx
x
x
xx
x
x
dy
yx
dx
dy
dx
dy
ee
y
x
e
dx
de y
xe
dx
ey
x
e
e
C
C
e
−−−
−
−
−−
−
=+→
−=→
=−
−
+ →
=− +
3.
You are thinking of buying a house and deciding whether you can afford to borrow $500,000.
You can
borrow the money at a 6% annual interest rate compounded continuously.
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 Spring '08
 Reshetiken
 Approximation

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