34.6.4Other Calculus Stuff

(ans.

homogeneous solution:

x'' + 4x' + 4x = 0

guess:

xh

= e

At

xh' = Ae

At

xh'' = A

2

At

e

xh = C1e–2t + C2te–2t

particular solution:

guess:

xp =

At + B

xp' = A

xp'' =

0

( 0) + 4( A) + 4( At + B) = 5t

( 4A + 4B) + ( 4A) t = ( 0) + ( 5) t

5

4( 1.25)

A =

4-- = 1.25

+ 4B = 0

B = –1.25

xp = 1.25t – 1.25

combine and solve for constants:

x( t) = x

h

+ x

p

= C

1

e–2t + C

2

te–2t + 1.25t – 1.25

for x(0) = 0

0 = C1e

–2( 0)

+ C2(

0) e

–2( 0)

+ 1.25( 0)

– 1.25 = C1 – 1.25

C1 = 1.25

for (d/dt)x(0) = 0

C2 = 1.25

x( t) = 1.25e2t + 1.25te2t + 1.25t – 1.25

d

a)

----( sin t + cos t)

dt

d

–2

b)

----

( ( t + 2) )

dt

d

( 5te

8t

c)

----

)

dt

a)

d

1

-----

-----------

dx

x + 1

b)

d

( e

–t

sin ( 2t – 4) )

----

dt

(ans.

d

1

–1

a)

-----

-----------

=

------------------

dx

( x + 1)

2

x + 1

b)

d

–t

–t

–t

----

( e sin ( 2t – 4) ) = –e sin ( 2t – 4)

+ 2e cos ( 2t – 4)

dt

(ans.

a)

∫e

2t

dt = 0.5e

2t

+ C

b)

1

+ -- sin 3θ + C

3

8. Solve the following differential equation.

x'' + 5x' + 3x = 3

x( 0) = 1

x'( 0) = 1

(ans.

x'' + 5x' + 3x = 3

x( 0) = 1

x'( 0) = 1

Homogeneous:

A2 + 5A + 4 = 0

– 5 ±

25 – 16

– 5 ± 3

= –4, –1

A = ----------------------------------- =

----------------

2

2

xh = C1e–4t + C2e–t

Particular:

xp = A

x'p = 0

x''p = 0

0 + 5( 0) + 3A = 3

A = 1

xp

= 1

Initial values:

x = xh + xp = C1e–4t + C2e–t + 1

1 = C1( 1) + C2( 1) + 1

C1 = –C2

x' = –4C1e–4t – C2e–t

1 = –4C1( 1) – C2( 1)

1

4( –C2) + C2 = –1

C2 =

3--

C

=

1

1

–--

1

–4t

1

–t

3

x = –3--e

+

3--e

+ 1

(ans.

∫

dV = ∫

a

V =

Adx

c – b

0

r =

b +

x

----------

a

2

c – b 2

2

c – b

c – b 2

2

A = π r = π b +

----------

= π b + 2π

----------

x + π

----------

x

a

x

a

a

a

2

c – b

c – b 2 2

V = ∫ π b + 2π

-----------

π

----------

x dx

a

x +

a

0

2

c – b 2 π

c – b

2

3

a

V =

π

b x + π

x

+

--

----------

x

a

3

a

0

V =

π

b

2

a + π

c – b

a

2

+

π

c – b

2

3

-----------

--

----------

a

a

3

a

2x'' + 4x' + 2x = 5

where, x( 0)

=

2

x'( 0)

=

0