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

Section 9.5 Linear Equations 625
Using Formula 98 in the Table of Integrals, we have
e 3t I − y 15e 3t sin 30t dt − 15 e 3t
s3 sin 30t 2 30 cos 30td 1 C
909
Since Is0d − 0, we get
I − 5
101 ssin 30t 2 10 cos 30td 1 Ce23t
2 50
101 1 C − 0
so Istd − 5
50
101 ssin 30t 2 10 cos 30td 1 101 e23t n
;
1–4 Determine whether the differential equation is linear.
1. y9 1 xsy − x 2 2. y9 2 x − y tan x
3. ue t − t 1 st
du
dt
5–14 Solve the differential equation.
4. dR
dt
1 t cos R − e 2t
5. y9 1 y − 1 6. y9 2 y − e x
7. y9 − x 2 y 8. 4x 3 y 1 x 4 y9 − sin 3 x
9. xy9 1 y − sx 10. 2xy9 1 y − 2sx
11. xy9 2 2y − x 2 , x . 0 12. y9 1 2xy − 1
13. t 2 dy
dt 1 3ty − s1 1 t 2 , t . 0
14. t ln t dr
dt 1 r − te t
15–20 Solve the initial-value problem.
15. x 2 y9 1 2xy − ln x, ys1d − 2
16. t dy 3 1 3t 2 y − cos t, ysd − 0
dt
17. t du − t 2 1 3u, t . 0, us2d − 4
dt
18. xy9 1 y − x ln x, ys1d − 0
19. xy9 − y 1 x 2 sin x, ysd − 0
20. sx 2 1 1d dy 1 3xsy 2 1d − 0, ys0d − 2
dx
21–22 Solve the differential equation and use a calcula tor to
graph several members of the family of solutions. How does
the solution curve change as C varies?
21. xy9 1 2y − e x 22. xy9 − x 2 1 2y
;
23. A Bernoulli differential equation (named after James
Bernoulli) is of the form
dy
dx
1 Psxdy − Qsxdy
n
Observe that, if n − 0 or 1, the Bernoulli equation is linear.
For other values of n, show that the substitution u − y 12n
transforms the Bernoulli equation into the linear equation
du
dx
1 s1 2 ndPsxdu − s1 2 ndQsxd
24–25 Use the method of Exercise 23 to solve the differential
equation.
24. xy9 1 y − 2xy 2
25. y9 1 2 x y − y 3
x 2
26. Solve the second-order equation xy0 1 2y9 − 12x 2 by
making the substitution u − y9.
27. In the circuit shown in Figure 4, a battery supplies a constant
voltage of 40 V, the inductance is 2 H, the resistance is
10 V, and Is0d − 0.
(a) Find Istd.
(b) Find the current after 0.1 seconds.
28. In the circuit shown in Figure 4, a generator supplies a voltage
of Estd − 40 sin 60t volts, the inductance is 1 H, the
resistance is 20 V, and Is0d − 1 A.
(a) Find Istd.
(b) Find the current after 0.1 seconds.
(c) Use a graphing device to draw the graph of the current
function.
29. 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
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