James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 9.3 Separable Equations 599
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CAS
21. Use Euler’s method with step size 0.5 to compute the
approximate y-values y 1, y 2, y 3, and y 4 of the solution of the
initial-value problem y9 − y 2 2x, ys1d − 0.
22. Use Euler’s method with step size 0.2 to estimate ys1d,
where ysxd is the solution of the initial-value problem
y9 − x 2 y 2 1 2 y2 , ys0d − 1.
23. Use Euler’s method with step size 0.1 to estimate ys0.5d,
where ysxd is the solution of the initial-value problem
y9 − y 1 xy, ys0d − 1.
24. (a) Use Euler’s method with step size 0.2 to estimate
ys0.6d, where ysxd is the solution of the initial-value
problem y9 − cossx 1 yd, ys0d − 0.
(b) Repeat part (a) with step size 0.1.
25. (a) Program a calculator or computer to use Euler’s method
to compute ys1d, where ysxd is the solution of the initialvalue
problem
dy
dx 1 3x 2 y − 6x 2 ys0d − 3
(i) h − 1 (ii) h − 0.1
(iii) h − 0.01 (iv) h − 0.001
(b) Verify that y − 2 1 e 2x 3 is the exact solution of the differential
equation.
(c) Find the errors in using Euler’s method to compute ys1d
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 ys2d, where y
is the solution of the initial-value problem
y9 − x 3 2 y 3 ys0d − 1
(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 C farads (F), and
a resistor with a resistance of R ohms (V). The voltage drop
across the capacitor is QyC, where Q is the charge (in coulombs,
C), so in this case Kirchhoff’s Law gives
But I − dQydt, so we have
R dQ
dt
RI 1 Q C − Estd
1 1 C Q − Estd
Suppose the resistance is 5 V, the capacitance is 0.05 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) Is there an equilibrium solution?
(d) If the initial charge is Qs0d − 0 C, use the direction field
to sketch the solution curve.
(e) If the initial charge is Qs0d − 0 C, use Euler’s method
with step size 0.1 to estimate the charge after half a
second.
E
C
28. In Exercise 9.1.14 we considered a 958C cup of coffee in a
208C room. Suppose it is known that the coffee cools at a rate
of 18C per minute when its temperature is 70°C.
(a) What does the differential equation become in this case?
(b) Sketch a direction field and use it to sketch the solution
curve for the initial-value problem. What is the limiting
value of the temperature?
(c) Use Euler’s method with step size h − 2 minutes to
estimate the temperature of the coffee after 10 minutes.
R
We have looked at first-order differential equations from a geometric point of view
(direction fields) and from a numerical point of view (Euler’s method). What about the
symbolic point of view? It would be nice to have an explicit formula for a solution of a
differential equation. Unfortunately, that is not always possible. But in this section we
examine a certain type of differential equation that can be solved explicitly.
A separable equation is a first-order differential equation in which the expression
for dyydx can be factored as a function of x times a function of y. In other words, it can
be written in the form
dy
− tsxd f syd
dx
The name separable comes from the fact that the expression on the right side can be “sep-
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