Tarea Aplicaciones De La Transformada 551t31

[CHAP. 3

APPLICATIONS TO DIFFERENTIAL EQUATIONS

102

31. Solve the partial differential equation

azy at2

16x + 20 sinx

subject to the conditions Y(O, t) = 0,

Y(-rr, t)

= 1671",

Yt (x, 0) = 0,

+

16x

Y(x, 0)

12 sin2x - 8 sin3x

Taking Laplace transforms, we find s2y -

8

Y(x, 0) -

d2y

Yt (x , 0) -

+

4 dxZ

16x

Y

8

+

20 sin x s

(1)

or, on using the given, conditions, d2y dx2 -

-4(s2 + 1)x

1

4 (sZ + 1)Y

5 sin x

=

y(O, s)

.

3s sm2x +

-- -

8

8

0,

y(,., s)

2s sin3x

16,.

=

(2) (3)

8

A particular solution of (2) has the form ax +

(4)

b sin x + c sin 2x + d sin 3x

Then substituting and equating co~fficients of like , we find the particular solution 16x + 8

20 sin x + 12s sin 2x ~ 88 sin 3x s(s2 + 5) 82 + 17 s2 + 37

(5)

The general solution of the equation (2) with right hand side replaced by zero [i.e. the complementary solution] is (6)

Yc

Thus the general solution of (2) is Y

=

Yp

+

(7)

Yc

Using the conditions (3) in (7), we find 0

= c2 = 0.

from which c1

Thus 16x

-8

y

+

20 sin x + 12s sin 2x 8(82+5) 82 +17

-

88 sin 3x 82+37

Then takin~ the inverse Laplace transform, we find the required solution

Y(x, t)

16x + 4 sin x (1- cos

V5 t)

+ 12 sin 2x cos Yf7 t -

8 sin 3x cos..J37 t

Supplementary Problems ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

Solve each of the following by using Laplace transforms and check solutions. 32. Y"(t)

+ 4Y(t)

33. Y"(t) - 3Y'(t)

Ans.

Y(t)

=

= 9t,

+

Y(O) = 0, Y'(O) = 7.

2Y(t) = 4t

+

12e-t,

An8.

Y(t)

=

Y(O) = 6, Y'(O) =· -1.

3et- 2e2t + 2t + 3 + 2e-t

3t + 2 sin 2t

CHAP. 3]

34.

35.

Y"(t)

Ans. 36.

=

Y(t)

=

Y(t)

+

25t2

+ Y(t) =

40t

125t2,

+

8 cost,

22

Y(O)

cost - 4 sin t

+

= Y'(O) = 0.

Y(O)

2e2t (2 sin t - 11 cost)

= 1, +

Y'(O)

= -1.

4t cost

= et, Y(O) = 0, Y'(O) = 0, Y"(O) = 0. . t = }tet + fs.e-lht { 9 cos t + -5vfa 2s m 2

Y'"(t) - Y(t)

Ans. 37.

+ 5Y(t) =

Y"(t) - 4Y'(t) Ans.

103

APPLICATIONS TO DIFFERENTIAL EQUATIONS

Y(t)

yiv(t)

+ 2Y"(t) + Y(t) =

Ans.

Y(t)

=

sin t,

V:

V3}

Y(O) = Y'(O)

= Y"(O) = Y"'(O) = 0.

i{(3- t2) sin t - 3t cost}

38. Find the general solution of the differential equations of: (a.) Problem 2, Page 82; (b) Problem 3, Page 83; (c) Problem 6, Page 84.

Ans.

c1 et + c2e2t + 4te2t e- t (c1 sin 2t. + c 2 cos 2t) + ie-t sin t

(a) Y

=

(b) Y

+ 9Y(t)

39.

Solve

Y"(t)

40.

Solve

yiv(t) - 16Y(t)

Y

Ans. 41.

Solve

Ans. 42.

=

= 30 sin t

if Y(O)

=

+ 3Y

Y(11-/2) = 0.

= 0,

Y'(O)

c1 sin 3t + c2 cos 3t +

Ans. Y(t)

= 2,

Y"(7r)

= 0,

=

2t

Y'"(7r)

+

7T

cos 2t

sin 3t

= -18.

tit

+

!et - fe3t

(e3u- eu) F(t- u) du

Solve the differential equation

where

Ans.

F(t) .

={

1

0

< t >

0

Y(t)

= i

sin 2t

and

Y(t)

= i

+

+ 4Y

F(t),

Y(O)

Y'(O)

=1

for t > 1

!-{cos (2t-- 2) - cos 2t}

sin 2t

= 0,

1 1

+

!(1 - cos 2t)

for t

<

1

Solve Problem 42 if: . (a.) F(t) = 1l(t - 2), [Heaviside's unit step function]; delta function]; (c) F(t) = 8(t - 2).

Ans.

(a.) Y(t) (b) Y(t) (c)

Y(t)

= i sin 2t if t < 2, t = sin 2t, t > 0

=t

sin 2t if t

sin 2t

+

(b) F(t)

= 8(t),

!{1 - cos (2t- 4)} if t > 2

< 2, !{sin 2t + sin (2t- 4)}2 if t > 2

ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

Solve each of the following by, using Laplace transforms and check solutions. 44.

t

= 0.

= F(t) if Y(O) = 1, Y'(O)

Y"

43.

= 0,

=

2(sin 2t- sin t)

Y" - 4Y' Y

= 18t if Y(O)

(c) Y

Y"

+ tY' -

Y

=

0,

Y(O)

=

45. tY"

+

(1- 2t)Y' - 2Y

46. tY"

+

(t - 1)Y' - Y ·= 0,

= 0,

0,

Y'(O)

= 1.

Ans. Y

Y(O) = 1, Y'(O) = 2.

=t Ans. Y = e2t

Y(O) = 5, Y(oo) = 0.

Ans. Y = 5e - t

47. Find the bounded solution of the equation

t2Y" which is such that Y(1)

= 2.

+

tY'

+

(t2- 1)Y

Ans. 2J 1 (t)/J1 (1)

=

0

[Dirac

1()4

[CHAP. 3

APPLICATIONS TO JJIFFERENTIAL EQUATIONS

SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS

Ans.

49.

50.

Solve

=

Y

t e-t

2 + -ft2 + -fe-t -!sin t +

=

=

=

if X(O)

Ans. X= 1+e-t-e-at-e-l>t, Y 51.

tY + Z { Y'- Z

Ans.

Y

54.

=

Z

(t- 1)e-t

e-t

=

1 - -j-e-t +

= -2, i

= 0.

Z(O)

sin t -

l

cost

=

Z

fl-e-t - !e2t + tte-t

= Y(O) = Y'(O) = 0 . = 0,

Y(O)

Y'(,.)

= 1,

given that Y(O)

where a= !(2-\1'2), b

Z(O)

= 1,

= f(2+\1'2)

= 0.

Z(O)

= -1.

= -1,

Y'(O)

= -J1 (t)- e-t

=

Y

Y'(O)

= Y'(O) = Z(O) = 0.

sin t + tte-t,

=

-3Y" + 3Z" te-t - 3 cost { tY" - Z' sin t

53. Solve

Ans.

= J0(t),

+ tZ' =

=

Z

cost,

= 1+e-t-oe-at-ae-bt

Solve Problem 49 with the conditions

5Z. Solve

i

fl-e-t + 4\e2t - !-cost-

X' + 2Y" e-t .{ X'+2X- Y 1

l

if Y(O)

=

Y

= 3,

subject to the conditions Y(O)

Y'- Z'- 2Y + 2Z . = sin t { Y" + 2Z' + Y 0 ·

Solve

Ans.

= =

Y' + Z' { Y"- z

48. Solve

.ft2 +it-

i- te - t,

Z

=

given that

Y(O)

= 2,

Z(O)

= 4,

Z"(O) = 0.

.ft2 + J+ te - t + tte-t +cost

Find the general solution of the system of equations in Problem 49.

Ans.

Y Z

c1 + c2 sin t + c3 cost + -ft2 + -fe-t

= 1 - c2 sin t - c3 cos t - -fe-t

APPLICATIONS TO MECHANICS 55• .'Referring to Fig. 3-1, Page 79, suppose that mass m has a force "f'(t), t

> 0 acting on it but that no

damping forces are present. (a) Show that if the mass starts from rest at a distance X= a from the equilibrium position (X then the displacement X at any time t > 0 can be .determined from the equation of motion

mX" + kX

=

"F(t),

X(O)

= a,

X'(O)

= 0),

=0

where primes denote derivatives with respect to t.

= F 0 (a constant) for t "f'(t) = F 0 e-at where a > 0.

(b) Find X at any time if "f'(t) (c) Find X at any time if

Ans.

(b)

X=

(c)

X

=

> 0.

F ( 1-cos'\jmt . [k ) a+k 0

Fo -~ a+ ma2 +k(e-at-cosyk/mt)

+

aF0 ym/k . -~ ma2 +k smyk/mt

56. Work Problem 55 if "ji'(t) = F 0 sin wt, treating the two cases: the physical significance of each case.

(a) w- ~ yk/m, (b) w

= yk/m.

Discuss

57. A particle moves along a line so that its displacement X from a fixed point 0 at any time t is given by X"(t) + 4 X(t) + 5 X(t)

=

80 sin 5t