1
00:00:01,570 --> 00:00:02,590
PROFESSOR: OK.

2
00:00:02,590 --> 00:00:07,790
Finally, I'm going to solve this
first order linear differential

3
00:00:07,790 --> 00:00:15,490
equation with a formula that
works for any source term.

4
00:00:15,490 --> 00:00:19,700
So we've solved it for specific,
nice, special source terms

5
00:00:19,700 --> 00:00:21,870
I'll remember later.

6
00:00:21,870 --> 00:00:24,430
But now we want a
formula for the solution

7
00:00:24,430 --> 00:00:27,130
to that equation, period.

8
00:00:27,130 --> 00:00:32,220
And we want to
understand the formula.

9
00:00:32,220 --> 00:00:34,680
So now, write the
formula down, and then

10
00:00:34,680 --> 00:00:37,140
let's see why it's right.

11
00:00:37,140 --> 00:00:39,610
And then of course, we could
put it into the equation

12
00:00:39,610 --> 00:00:42,000
and confirm that it's right.

13
00:00:42,000 --> 00:00:44,240
OK the formula is
going to be-- this

14
00:00:44,240 --> 00:00:49,140
is the big formula you
could say, for first order

15
00:00:49,140 --> 00:00:51,170
linear equations.

16
00:00:51,170 --> 00:00:52,740
So y of t.

17
00:00:52,740 --> 00:00:59,960
First we have the result
of the amount-- I'm

18
00:00:59,960 --> 00:01:04,040
thinking of this as balance.

19
00:01:04,040 --> 00:01:09,130
The money in the bank is why
it's increasing at this rate,

20
00:01:09,130 --> 00:01:11,840
because of interest being added.

21
00:01:11,840 --> 00:01:15,620
And it's increasing at this
rate because of new deposits

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00:01:15,620 --> 00:01:16,930
being added.

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00:01:16,930 --> 00:01:21,220
And oh, maybe I should
say about those deposits,

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00:01:21,220 --> 00:01:26,600
I'm not thinking of like deposit
once a year, or once a month,

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00:01:26,600 --> 00:01:30,170
or once a minute, even.

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00:01:30,170 --> 00:01:31,710
Deposit continuously.

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00:01:31,710 --> 00:01:33,990
We're talking differential
equations here.

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00:01:33,990 --> 00:01:37,770
Time is running all-- the
clock is always running.

29
00:01:37,770 --> 00:01:40,700
So you're--
theoretically, at least--

30
00:01:40,700 --> 00:01:46,490
you're depositing at a
rate of so much per second,

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00:01:46,490 --> 00:01:47,890
all the time.

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00:01:47,890 --> 00:01:49,740
OK.

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00:01:49,740 --> 00:01:54,380
So first, the y of 0,
that's a once in a lifetime

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00:01:54,380 --> 00:01:56,610
deposit to start the account.

35
00:01:56,610 --> 00:02:02,780
And it grows, as we know,
with the interest rate

36
00:02:02,780 --> 00:02:05,570
as the exponent.

37
00:02:05,570 --> 00:02:10,720
The question is-- so that's
the null solution that

38
00:02:10,720 --> 00:02:14,540
matches the initial condition.

39
00:02:14,540 --> 00:02:18,050
No solution, because there
are no deposits here.

40
00:02:18,050 --> 00:02:22,160
This solves this part of
the equation, the null part.

41
00:02:22,160 --> 00:02:26,140
But now I want to add in
a particular solution,

42
00:02:26,140 --> 00:02:28,100
and it'll be the
particular solution

43
00:02:28,100 --> 00:02:30,500
that matches the deposits.

44
00:02:30,500 --> 00:02:34,100
And I want to
understand this formula.

45
00:02:34,100 --> 00:02:35,780
So here's the formula.

46
00:02:35,780 --> 00:02:41,870
I'm going to-- the deposits
each deposit goes in at time T,

47
00:02:41,870 --> 00:02:46,000
goes in at some time,
and then it grows.

48
00:02:46,000 --> 00:02:48,050
Once you've made
the deposit, it's

49
00:02:48,050 --> 00:02:50,960
going to grow exponentially.

50
00:02:50,960 --> 00:02:53,980
With [INAUDIBLE], it will
grow over the remaining time.

51
00:02:53,980 --> 00:02:55,500
So here's the formula.

52
00:02:55,500 --> 00:03:02,340
You make deposits at any time,
s, between s equal 0 and s

53
00:03:02,340 --> 00:03:03,760
equal t.

54
00:03:03,760 --> 00:03:06,210
So s is the running clock.

55
00:03:06,210 --> 00:03:12,030
T is the-- we look at our
account and see what's in it.

56
00:03:12,030 --> 00:03:15,360
And this deposit
is made at time s,

57
00:03:15,360 --> 00:03:19,670
and then it grows over the
remaining time, from s to t.

58
00:03:19,670 --> 00:03:27,410
So it grows by a factor
e to the a, t minus s.

59
00:03:27,410 --> 00:03:29,190
That's the key.

60
00:03:29,190 --> 00:03:34,000
And now we add up all those
deposits with their growth.

61
00:03:34,000 --> 00:03:40,800
So that addition for continuous
time addition is integration.

62
00:03:40,800 --> 00:03:42,960
That's the whole
idea of the integral,

63
00:03:42,960 --> 00:03:46,360
is add up continuously.

64
00:03:46,360 --> 00:03:50,105
So there's my formula that
I'm hoping you will admire.

65
00:03:52,800 --> 00:03:58,290
The no solution that grows
out of the initial condition.

66
00:03:58,290 --> 00:04:06,040
The particular solution that
grows out of the source term,

67
00:04:06,040 --> 00:04:07,330
q.

68
00:04:07,330 --> 00:04:11,430
Of course I've used
q of t all the way.

69
00:04:11,430 --> 00:04:13,110
Here I call it q of s.

70
00:04:13,110 --> 00:04:15,980
I have to introduce a
integration variable

71
00:04:15,980 --> 00:04:21,019
s, which goes from the
start of these deposits

72
00:04:21,019 --> 00:04:24,960
until the current time,
and grows like that.

73
00:04:24,960 --> 00:04:27,020
So that's the formula.

74
00:04:27,020 --> 00:04:30,510
I could put a big box around it.

75
00:04:30,510 --> 00:04:34,110
Let me start a box anyway.

76
00:04:34,110 --> 00:04:36,840
I could check that it's correct.

77
00:04:36,840 --> 00:04:39,420
But I hope you see
why it's right.

78
00:04:39,420 --> 00:04:47,510
And let me make some
comment on the examples of q

79
00:04:47,510 --> 00:04:49,710
where we didn't
have this formula.

80
00:04:49,710 --> 00:04:52,010
We just went for it directly.

81
00:04:52,010 --> 00:04:56,220
So we started with,
these were special.

82
00:04:56,220 --> 00:05:00,340
Especially nice, you could say.

83
00:05:00,340 --> 00:05:05,910
Q of t equal a constant.

84
00:05:05,910 --> 00:05:08,860
That was the first video.

85
00:05:08,860 --> 00:05:14,500
Then we did q of t
equal an exponential.

86
00:05:14,500 --> 00:05:16,840
That was the second one.

87
00:05:16,840 --> 00:05:21,340
Then we did-- so and then we
found the exponential response.

88
00:05:21,340 --> 00:05:24,800
Then I did oscillation.

89
00:05:24,800 --> 00:05:33,610
Cosine-- well, let me put it
over here-- cosine of omega t.

90
00:05:33,610 --> 00:05:35,830
Or plus sine of omega t.

91
00:05:35,830 --> 00:05:38,210
You remember, those two kind
of had to come together.

92
00:05:38,210 --> 00:05:41,220
We couldn't stick
with cosines alone,

93
00:05:41,220 --> 00:05:43,980
because the
derivative of a cosine

94
00:05:43,980 --> 00:05:48,430
here would give us a sign, so
signs got into the picture.

95
00:05:48,430 --> 00:05:49,965
So we found the
formula for that,

96
00:05:49,965 --> 00:05:51,310
that took a little more work.

97
00:05:51,310 --> 00:05:55,570
And in fact we did it by a real
method and a complex method.

98
00:05:55,570 --> 00:05:59,170
And, now are there any other
nice functions in calculus?

99
00:05:59,170 --> 00:06:04,100
Well the next video,
I'm going to tell you

100
00:06:04,100 --> 00:06:08,290
about two more functions
that I think are very nice.

101
00:06:08,290 --> 00:06:09,185
Step function.

102
00:06:12,530 --> 00:06:16,370
So the deposit-- we don't
make any deposits up

103
00:06:16,370 --> 00:06:18,780
until some time,
and then we start,

104
00:06:18,780 --> 00:06:21,850
then we go change to constant.

105
00:06:21,850 --> 00:06:25,740
So the step function is
0 and then a constant.

106
00:06:25,740 --> 00:06:31,630
And also-- and this is the
especially interesting one--

107
00:06:31,630 --> 00:06:34,560
a delta function.

108
00:06:34,560 --> 00:06:37,350
And what is a delta function?

109
00:06:37,350 --> 00:06:41,540
Which doesn't always come
into the basic differential

110
00:06:41,540 --> 00:06:44,000
equations, course,
but it belongs there.

111
00:06:44,000 --> 00:06:48,540
Because in a model of
reality, a delta function

112
00:06:48,540 --> 00:06:51,070
is like a golf club
hitting a golf ball.

113
00:06:51,070 --> 00:06:53,340
In an instant
something happened.

114
00:06:53,340 --> 00:06:56,730
Or a baseball bat
hitting a baseball.

115
00:06:56,730 --> 00:06:59,730
It gives it an instant velocity.

116
00:06:59,730 --> 00:07:01,730
It's an impulse.

117
00:07:01,730 --> 00:07:07,310
So step function is like a
light switch off and then on.

118
00:07:07,310 --> 00:07:11,680
Delta function is all in
one instant, an impulse,

119
00:07:11,680 --> 00:07:13,620
and you'll see that coming.

120
00:07:13,620 --> 00:07:17,800
And then those are
the-- oh, and maybe I

121
00:07:17,800 --> 00:07:22,080
could include-- let's see,
if I've got a constant,

122
00:07:22,080 --> 00:07:26,410
maybe should include
t, t squared, so on.

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00:07:26,410 --> 00:07:28,830
Powers of t are not too bad.

124
00:07:28,830 --> 00:07:29,780
We could get--

125
00:07:29,780 --> 00:07:34,610
So for all of these special ones
that I call the nice functions.

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00:07:34,610 --> 00:07:39,190
For all of those-- and we could
multiply t times e to the st,

127
00:07:39,190 --> 00:07:40,360
that would still be nice.

128
00:07:40,360 --> 00:07:42,520
And we could have
a simple formula.

129
00:07:42,520 --> 00:07:47,080
Those are the ones that give
simple, direct, interesting,

130
00:07:47,080 --> 00:07:49,050
answers in themselves.

131
00:07:49,050 --> 00:07:52,960
And this is the one that
gives the general answer.

132
00:07:52,960 --> 00:07:59,650
And if I put q be any one of
those in this general formula,

133
00:07:59,650 --> 00:08:02,400
I'll get the special one.

134
00:08:02,400 --> 00:08:06,370
You can see how if I
put a constant in there,

135
00:08:06,370 --> 00:08:09,960
that will be exactly
what we found

136
00:08:09,960 --> 00:08:15,980
at the very, very beginning
for the response to a constant.

137
00:08:15,980 --> 00:08:16,590
Right.

138
00:08:16,590 --> 00:08:18,765
So this is the
general expression.

139
00:08:21,800 --> 00:08:25,890
I feel I should
say more about it.

140
00:08:25,890 --> 00:08:29,550
I guess I should say, two
more things I want to add

141
00:08:29,550 --> 00:08:31,240
about this general formula.

142
00:08:31,240 --> 00:08:34,610
One is, I should check
that it's correct.

143
00:08:34,610 --> 00:08:37,450
But I hope you saw,
it had to be right.

144
00:08:37,450 --> 00:08:42,059
This input went in, it
grew, everything was linear.

145
00:08:42,059 --> 00:08:46,390
So I could just add the separate
growth, separate results,

146
00:08:46,390 --> 00:08:52,780
to find out what the balance
was at the final time, t.

147
00:08:52,780 --> 00:08:54,810
But still, I can check it.

148
00:08:54,810 --> 00:08:56,330
And I can derive it.

149
00:08:59,580 --> 00:09:03,770
That step is always
often done by what's

150
00:09:03,770 --> 00:09:06,930
called an integrating factor.

151
00:09:06,930 --> 00:09:09,480
So there will be
a quick video that

152
00:09:09,480 --> 00:09:14,950
shows you how an integrating
factor leads to that formula.

153
00:09:14,950 --> 00:09:18,690
Right in this video,
I'm just saying I

154
00:09:18,690 --> 00:09:22,080
was led to it by common sense.

155
00:09:22,080 --> 00:09:23,800
But I can check
that it's correct.

156
00:09:23,800 --> 00:09:26,060
So let me check
that it's correct.

157
00:09:26,060 --> 00:09:26,620
OK.

158
00:09:26,620 --> 00:09:28,460
This part I can see is correct.

159
00:09:28,460 --> 00:09:30,470
So I'd like to
work with that one

160
00:09:30,470 --> 00:09:33,240
and show that that is
a particular solution

161
00:09:33,240 --> 00:09:34,750
to the differential equation.

162
00:09:34,750 --> 00:09:36,100
Can I do that?

163
00:09:36,100 --> 00:09:44,070
I want to show that
this-- I'm looking now

164
00:09:44,070 --> 00:09:51,155
at this y particular, and I'll
factor out the e to the at,

165
00:09:51,155 --> 00:09:53,200
because it doesn't depend on s.

166
00:09:53,200 --> 00:09:55,730
It's not involved
in the integration.

167
00:09:55,730 --> 00:10:01,990
So this is in the integral from
s equals 0 to s equal t of e

168
00:10:01,990 --> 00:10:04,730
to the minus as,
that has an s in it.

169
00:10:04,730 --> 00:10:08,150
Q of s, ds.

170
00:10:08,150 --> 00:10:11,980
This is all the stuff
that depends on what time

171
00:10:11,980 --> 00:10:14,630
the deposit was made, time s.

172
00:10:14,630 --> 00:10:19,380
And what time we're looking at
the balance, the later time, t.

173
00:10:19,380 --> 00:10:21,350
OK.

174
00:10:21,350 --> 00:10:25,670
This is a product of one
term times another term.

175
00:10:25,670 --> 00:10:29,300
And when I put it into
the differential equation,

176
00:10:29,300 --> 00:10:31,400
I'll use the product rule.

177
00:10:31,400 --> 00:10:33,900
So the derivative
of that product

178
00:10:33,900 --> 00:10:36,690
will have two terms
from the product rule.

179
00:10:36,690 --> 00:10:39,560
And those two terms
will be-- if all

180
00:10:39,560 --> 00:10:42,580
goes well-- will be the two
terms in the differential

181
00:10:42,580 --> 00:10:43,580
equation.

182
00:10:43,580 --> 00:10:47,140
So can I take the derivative
of this by the product rule?

183
00:10:47,140 --> 00:10:49,990
So I take the derivative
of the first thing,

184
00:10:49,990 --> 00:11:00,730
so now I'm computing dy,
dt, by ordinary calculus.

185
00:11:00,730 --> 00:11:08,610
So the derivative of that is a
e the at times the second term.

186
00:11:08,610 --> 00:11:11,580
That's just ay.

187
00:11:11,580 --> 00:11:14,310
That's a times the y
that we had before.

188
00:11:14,310 --> 00:11:18,080
Because the derivative of e
to the at, by the chain rule,

189
00:11:18,080 --> 00:11:20,440
brings down an a,
and we have that.

190
00:11:20,440 --> 00:11:23,610
But now the product
rule says also I

191
00:11:23,610 --> 00:11:32,040
must take that term times
the derivative of this term.

192
00:11:32,040 --> 00:11:34,260
And that looks messier.

193
00:11:34,260 --> 00:11:39,740
What's the derivative
of that function?

194
00:11:39,740 --> 00:11:42,290
It's a function of t.

195
00:11:42,290 --> 00:11:44,430
But it's an integral.

196
00:11:44,430 --> 00:11:45,480
It's a function of t.

197
00:11:45,480 --> 00:11:47,920
What's it's time derivative?

198
00:11:47,920 --> 00:11:51,760
That's the last piece
of the product rule.

199
00:11:51,760 --> 00:11:54,460
Well, look what it is.

200
00:11:54,460 --> 00:11:58,440
It's the integral up to
time t of some function.

201
00:11:58,440 --> 00:12:00,410
And the fundamental
theorem of calculus

202
00:12:00,410 --> 00:12:05,380
says that the derivative
of the integral

203
00:12:05,380 --> 00:12:08,320
is the original function, right?

204
00:12:08,320 --> 00:12:12,780
So the derivative
of this integral

205
00:12:12,780 --> 00:12:16,870
is the original function
that we integrated at time t.

206
00:12:16,870 --> 00:12:20,580
So it's e to the
minus a t cubed t.

207
00:12:26,570 --> 00:12:29,870
That's the second term
from the product rule.

208
00:12:29,870 --> 00:12:34,540
And OK, this was
the ay, perfectly.

209
00:12:34,540 --> 00:12:36,030
And what do I have here?

210
00:12:36,030 --> 00:12:39,070
E to the at cancels
e to the minus at,

211
00:12:39,070 --> 00:12:40,980
and it's the source term q of t.

212
00:12:40,980 --> 00:12:46,540
So I have ay plus q of t,
the correct right hand side

213
00:12:46,540 --> 00:12:48,510
for the differential equation.

214
00:12:48,510 --> 00:12:55,320
So this, that formula, let
me bring it down once more,

215
00:12:55,320 --> 00:13:02,740
is sort of the climax of
this beginning of the videos

216
00:13:02,740 --> 00:13:06,740
to see first order linear
differential equations.

217
00:13:06,740 --> 00:13:10,130
So what's coming now are
step and delta function.

218
00:13:10,130 --> 00:13:12,930
I just want to
speak about those.

219
00:13:12,930 --> 00:13:15,580
And that takes a few
minutes separately.

220
00:13:15,580 --> 00:13:21,810
And then another step will
be to allow the interest

221
00:13:21,810 --> 00:13:23,810
rate to change.

222
00:13:23,810 --> 00:13:26,940
We made our problem
simple because we've

223
00:13:26,940 --> 00:13:29,110
kept a constant interest rate.

224
00:13:29,110 --> 00:13:31,350
So I'll let the
interest rate change.

225
00:13:31,350 --> 00:13:34,380
And then after that
comes the real step

226
00:13:34,380 --> 00:13:36,900
to non-linear equations.

227
00:13:36,900 --> 00:13:41,090
So delta functions,
varying interest rate,

228
00:13:41,090 --> 00:13:46,080
and then non-linear equations.

229
00:13:46,080 --> 00:13:48,290
And then on to second
order equations

230
00:13:48,290 --> 00:13:51,800
and all the rest of the theory
of differential equations.

231
00:13:51,800 --> 00:13:52,300
Good.

232
00:13:52,300 --> 00:13:54,130
Thank you.