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22.
Use Euler’s method with step size
to estimate
, where
is the solution of the initialvalue problem
,
.
Use Euler’s method with step size
to estimate
, where
is the solution of the initialvalue problem
,
.
24.
(a) Use Euler’s method with step size
to estimate
,
where
is the solution of the initialvalue problem
,.
(b) Repeat part (a) with step size
.
;
25.
(a) Program a calculator or computer to use Euler’s method to
compute
, where
is the solution of the initialvalue
problem
(i)
(ii)
(iii)
(iv)
(b) Verify that
is the exact solution of the differ
ential equation.
(c) Find the errors in using Euler’s method to compute
with the step sizes in part (a). What happens to the error
when the step size is divided by 10?
26.
(a) Program your computer algebra system, using Euler’s
method with step size 0.01, to calculate
, where
is the solution of the initialvalue problem
(b) Check your work by using the CAS to draw the solution
curve.
27.
The ﬁgure shows a circuit containing an electromotive force, a
capacitor with a capacitance of
farads (F), and a resistor with
a resistance of
ohms (
). The voltage drop across the capaci
tor is
, where
is the charge (in coulombs), so in this case
Kirchhoff’s Law gives
But
, so we have
Suppose the resistance is
, the capacitance is
F, and a
battery gives a constant voltage of 60 V.
(a) Draw a direction ﬁeld for this differential equation.
(b) What is the limiting value of the charge?
C
E
R
0.05
±
5
R
dQ
dt
²
1
C
Q
±
E
±
t
²
I
±
dQ
³
dt
RI
²
Q
C
±
E
±
t
²
Q
Q
³
C
±
R
C
y
±
0
²
±
1
y
³
±
x
3
´
y
3
y
y
±
2
²
CAS
y
±
1
²
y
±
2
²
e
´
x
3
h
±
0.001
h
±
0.01
h
±
0.1
h
±
1
y
±
0
²
±
3
dy
dx
²
3
x
2
y
±
6
x
2
y
±
x
²
y
±
1
²
0.1
y
±
1
²
±
0
y
³
±
x
´
xy
y
±
x
²
y
±
1.4
²
0.2
y
±
0
²
±
1
y
³
±
y
²
y
±
x
²
y
±
0.5
²
0.1
23.
y
±
0
²
±
0
y
³
±
1
´
y
±
x
²
y
±
1
²
0.2
Make a rough sketch of a direction ﬁeld for the autonomous
differential equation
, where the graph of
is as
shown. How does the limiting behavior of solutions depend
on the value of
?
(a) Use Euler’s method with each of the following step sizes to
estimate the value of
, where
is the solution of the
initialvalue problem
.
(i)
(ii)
(iii)
(b) We know that the exact solution of the initialvalue
problem in part (a) is
. Draw, as accurately as you
can, the graph of
, together with the
Euler approximations using the step sizes in part (a).
(Your sketches should resemble Figures 12, 13, and 14.)
Use your sketches to decide whether your estimates in
part (a) are underestimates or overestimates.
(c) The error in Euler’s method is the difference between
the exact value and the approximate value. Find the errors
made in part (a) in using Euler’s method to estimate the
true value of
, namely
. What happens to the error
each time the step size is halved?
20.
A direction ﬁeld for a differential equation is shown. Draw,
with a ruler, the graphs of the Euler approximations to the
solution curve that passes through the origin. Use step sizes
and
. Will the Euler estimates be underestimates
or overestimates? Explain.
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 Fall '07
 Temkin
 Differential Equations, Calculus, Equations

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