1
00:00:00,000 --> 00:00:00,990


2
00:00:00,990 --> 00:00:04,030
Welcome to the video on the introduction to

3
00:00:04,030 --> 00:00:05,020
differential equations.

4
00:00:05,019 --> 00:00:06,199
What's a differential?

5
00:00:06,200 --> 00:00:10,699
Well, we've been doing things like if I said that y is

6
00:00:10,699 --> 00:00:12,899
equal to, I don't know, 3x.

7
00:00:12,900 --> 00:00:15,496
If we took the derivative of both sides of this equation and

8
00:00:15,496 --> 00:00:18,719
we said -- one of the notations we used is we said dy

9
00:00:18,719 --> 00:00:21,619
over dx is equal to 3.

10
00:00:21,620 --> 00:00:23,830
We took the derivative of both sides of that.

11
00:00:23,829 --> 00:00:26,459
But this you could almost use differential notation.

12
00:00:26,460 --> 00:00:31,640
dy is a differential and dx is a differential.

13
00:00:31,640 --> 00:00:33,730
What does a differential mean?

14
00:00:33,729 --> 00:00:38,169
A differential just means an infinitely small change in y,

15
00:00:38,170 --> 00:00:41,140
in this case, so that's a differential in y, or an

16
00:00:41,140 --> 00:00:42,329
infinitely small change in x.

17
00:00:42,329 --> 00:00:46,570
So a differential it's like a difference -- when we learned

18
00:00:46,570 --> 00:00:51,340
slope, we said slope is change in y over change in x.

19
00:00:51,340 --> 00:00:53,950
Or you could say the difference in y over difference in x.

20
00:00:53,950 --> 00:00:57,100
But once we started taking the slope of curves we had to make

21
00:00:57,100 --> 00:01:01,130
this difference and this difference approach 0.

22
00:01:01,130 --> 00:01:02,800
That's what a differential essentially is.

23
00:01:02,799 --> 00:01:04,899
A differential is a really, really, really, really

24
00:01:04,900 --> 00:01:06,810
infinitely small difference.

25
00:01:06,810 --> 00:01:08,909
So what is a differential equation?

26
00:01:08,909 --> 00:01:11,679
Well it's an equation that involves differentials.

27
00:01:11,680 --> 00:01:14,460
So, for example, this is a differential equation.

28
00:01:14,459 --> 00:01:18,069
This will be the first example that we solve.

29
00:01:18,069 --> 00:01:26,789
dy over dx is equal to x squared plus 1.

30
00:01:26,790 --> 00:01:29,520
So how do we solve this differential equation?

31
00:01:29,519 --> 00:01:31,750
Well, there's a couple of ways you could say it.

32
00:01:31,750 --> 00:01:35,030
You say oh, well this is the same thing as f prime of x.

33
00:01:35,030 --> 00:01:36,859
You know, y to the function of x and you just take the

34
00:01:36,859 --> 00:01:38,530
antiderivative of both sides.

35
00:01:38,530 --> 00:01:40,700
But let's solve this properly as a differential equation.

36
00:01:40,700 --> 00:01:44,329
So you can actually manipulate these differentials the same

37
00:01:44,329 --> 00:01:47,780
way that you would manipulate numbers for the most part.

38
00:01:47,780 --> 00:01:49,030
And so what could we do here?

39
00:01:49,030 --> 00:01:52,840
Let's multiply both sides of this equation by dx.

40
00:01:52,840 --> 00:02:00,510
So you get, for a very small change in y, dy, you get x

41
00:02:00,510 --> 00:02:05,550
squared plus 1 times a very small change in the x.

42
00:02:05,549 --> 00:02:08,500
Or another way to say it is for a very small change in x, if

43
00:02:08,500 --> 00:02:12,330
you want to figure out how much does y change, you multiply

44
00:02:12,330 --> 00:02:15,760
times x squared plus 1 wherever you are on the curve.

45
00:02:15,759 --> 00:02:17,189
So what does that do for us?

46
00:02:17,189 --> 00:02:20,240
Well let's take the antiderivative of both

47
00:02:20,240 --> 00:02:21,810
sides or let's take the integral of both sides.

48
00:02:21,810 --> 00:02:25,890
And if you watched all of the videos on integration and the

49
00:02:25,889 --> 00:02:30,589
definite integral and area and a curve, you realize that an

50
00:02:30,590 --> 00:02:34,950
integral is essentially a sum, it's kind of an infinite

51
00:02:34,949 --> 00:02:38,399
sum of a bunch of these infinitely small dy's.

52
00:02:38,400 --> 00:02:43,020
So if you take the sum of all of the changes in dy's,

53
00:02:43,020 --> 00:02:45,070
you're left with a y on this side of this equation.

54
00:02:45,069 --> 00:02:46,620
You might want to re-watch all the videos that we

55
00:02:46,620 --> 00:02:49,009
did on integration.

56
00:02:49,009 --> 00:02:49,759
Then what is this?

57
00:02:49,759 --> 00:02:52,669
This is essentially just -- this is the indefinite

58
00:02:52,669 --> 00:02:54,229
integral of x squared plus 1.

59
00:02:54,229 --> 00:02:56,019
So it's the antiderivative of that.

60
00:02:56,020 --> 00:02:59,510
And what's the antiderivative of x squared plus 1?

61
00:02:59,509 --> 00:03:04,500
Well it's x to the third over 3.

62
00:03:04,500 --> 00:03:06,689
We're just taking the derivative backwards.

63
00:03:06,689 --> 00:03:09,689
And once again, watch the videos on the antiderivative.

64
00:03:09,689 --> 00:03:12,030
There's I think eight or nine of them.

65
00:03:12,030 --> 00:03:15,120
Plus x and then plus c.

66
00:03:15,120 --> 00:03:16,890
And where does the c come from?

67
00:03:16,889 --> 00:03:18,759
Well we know that when we take the derivative of a

68
00:03:18,759 --> 00:03:20,590
constant it goes to 0.

69
00:03:20,590 --> 00:03:22,539
So when you take the antiderivative, we're like

70
00:03:22,539 --> 00:03:24,139
well, there could have been a constant there.

71
00:03:24,139 --> 00:03:26,289
And that's where that plus c comes from.

72
00:03:26,289 --> 00:03:33,530
So this is the general solution to this differential equation.

73
00:03:33,530 --> 00:03:34,770
And that's something interesting.

74
00:03:34,770 --> 00:03:37,909
So with traditional equations, the solution tended to

75
00:03:37,909 --> 00:03:39,680
be a number, right?

76
00:03:39,680 --> 00:03:46,920
If I just told you y is equal to 2y minus 1.

77
00:03:46,919 --> 00:03:49,819
Then you would get minus y is equal to minus 1, you'd

78
00:03:49,819 --> 00:03:51,379
get y is equal to 1.

79
00:03:51,379 --> 00:03:54,389
So this is a traditional equation and your solution

80
00:03:54,389 --> 00:03:58,359
was just a value -- you solved for the variable.

81
00:03:58,360 --> 00:04:00,370
Differential equations are something different.

82
00:04:00,370 --> 00:04:02,689
The solution is actually a function.

83
00:04:02,689 --> 00:04:06,469
You're saying what function satisfies this

84
00:04:06,469 --> 00:04:07,379
differential equation.

85
00:04:07,379 --> 00:04:08,909
So that's something to keep in mind.

86
00:04:08,909 --> 00:04:10,669
Right now we're doing very basic differential equations,

87
00:04:10,669 --> 00:04:12,489
but that's something to keep in mind the whole time you learn

88
00:04:12,490 --> 00:04:13,210
differential equations.

89
00:04:13,210 --> 00:04:17,519
I think I'll eventually do a play list on essentially

90
00:04:17,519 --> 00:04:19,599
an introductory course on differential equations that

91
00:04:19,600 --> 00:04:21,000
you would take at college.

92
00:04:21,000 --> 00:04:22,879
And that applies even when you start doing partial

93
00:04:22,879 --> 00:04:25,089
differential equations, et cetera, et cetera.

94
00:04:25,089 --> 00:04:28,729
The solution to a differential equation is not a number,

95
00:04:28,730 --> 00:04:30,660
it is a function.

96
00:04:30,660 --> 00:04:32,490
So anyway, this was the general solution to this

97
00:04:32,490 --> 00:04:33,490
differential equation.

98
00:04:33,490 --> 00:04:36,370
And if you want the particular solution, people normally give

99
00:04:36,370 --> 00:04:42,120
you initial conditions or they give you points on the function

100
00:04:42,120 --> 00:04:43,740
and then you can substitute back.

101
00:04:43,740 --> 00:04:48,150
In this problem they said that dy dx equals x squared plus 1.

102
00:04:48,149 --> 00:04:51,229
And they said that -- let me switch colors -- they said that

103
00:04:51,230 --> 00:04:56,980
y is equal to 1 at x is equal to 1, or y is equal to 1

104
00:04:56,980 --> 00:04:58,689
when x is equal to 1.

105
00:04:58,689 --> 00:05:01,379
So we can use this information now to solve for c.

106
00:05:01,379 --> 00:05:02,079
How do you do that?

107
00:05:02,079 --> 00:05:05,060
Well it says y is equal to 1.

108
00:05:05,060 --> 00:05:10,439
So 1 is equal to when x is equal to 1, is equal to 1/3,

109
00:05:10,439 --> 00:05:18,170
right, 1 over 3, 1 to the third power over 3, plus 1 plus c.

110
00:05:18,170 --> 00:05:19,129
And let's see what I can do.

111
00:05:19,129 --> 00:05:22,199
Subtract one from both sides -- this comes 0.

112
00:05:22,199 --> 00:05:24,909
Subtract 1/3 from both sides and you get c is

113
00:05:24,910 --> 00:05:27,420
equal to minus 1/3.

114
00:05:27,420 --> 00:05:31,860
So using these conditions, a point where this function

115
00:05:31,860 --> 00:05:34,560
crosses through, we can now give you the particular

116
00:05:34,560 --> 00:05:36,959
solution to this differential equation.

117
00:05:36,959 --> 00:05:48,009
And that is y is equal to x to the third over 3 plus x minus

118
00:05:48,009 --> 00:05:50,279
1/3, and then we just solve for the c.

119
00:05:50,279 --> 00:05:54,669
And if you don't believe me, take this expression and

120
00:05:54,670 --> 00:05:57,759
substitute it here and you will see that it equals, if you were

121
00:05:57,759 --> 00:06:00,310
to take the derivative of y with respect to x, you would

122
00:06:00,310 --> 00:06:02,670
see that it equals x squared plus 1.

123
00:06:02,670 --> 00:06:03,550
Let's do another one.

124
00:06:03,550 --> 00:06:08,410


125
00:06:08,410 --> 00:06:11,420
So it tells us -- so this is a little bit more interesting --

126
00:06:11,420 --> 00:06:24,160
it says dy dx is equal to x over y, and it has

127
00:06:24,160 --> 00:06:24,830
the same conditions.

128
00:06:24,829 --> 00:06:29,979
It says y equals 1 at x equals 1.

129
00:06:29,980 --> 00:06:31,640
Or when x equals 1, y equals 1.

130
00:06:31,639 --> 00:06:34,789
So let's find the function that satisfies this equation.

131
00:06:34,790 --> 00:06:37,129
This one's interesting because we have an x and a y on

132
00:06:37,129 --> 00:06:38,379
the right hand side of the equation.

133
00:06:38,379 --> 00:06:40,909
So this kind of looks like something we got after we

134
00:06:40,910 --> 00:06:42,680
did some type of implicit differentiation.

135
00:06:42,680 --> 00:06:45,480
But let's see where we go.

136
00:06:45,480 --> 00:06:46,840
So let's do the same thing.

137
00:06:46,839 --> 00:06:51,279
Let's multiply both sides of this equation by dx until you

138
00:06:51,279 --> 00:06:57,379
get dy is equal to x over y dx.

139
00:06:57,379 --> 00:07:01,290
And let's get this y over onto the dy side, because

140
00:07:01,290 --> 00:07:03,280
it will be easy to take the antiderivative then.

141
00:07:03,279 --> 00:07:11,989
So we get y dy is equal to x dx.

142
00:07:11,990 --> 00:07:15,795
Now we can take the integral of both sides of this equation, or

143
00:07:15,795 --> 00:07:18,069
take the antiderivative on this side with respect to y on

144
00:07:18,069 --> 00:07:19,959
the side with respect to x.

145
00:07:19,959 --> 00:07:22,629


146
00:07:22,629 --> 00:07:24,790
I don't know why I did it in brown.

147
00:07:24,790 --> 00:07:27,650
So what's the antiderivative here?

148
00:07:27,649 --> 00:07:34,500
Well, it is y squared over 2.

149
00:07:34,500 --> 00:07:38,060
And I could say plus c, some constant.

150
00:07:38,060 --> 00:07:39,180
Let me do that.

151
00:07:39,180 --> 00:07:40,750
There could be a constant here.

152
00:07:40,750 --> 00:07:44,410
I'll call it c1.

153
00:07:44,410 --> 00:07:46,410
We don't know what that constant is.

154
00:07:46,410 --> 00:07:57,590
And that equals x squared over 2 plus c2, right?

155
00:07:57,589 --> 00:07:58,719
Or some other constant.

156
00:07:58,720 --> 00:08:00,630
Maybe it's the same number, I don't know.

157
00:08:00,629 --> 00:08:02,350
But we don't know what either of these are.

158
00:08:02,350 --> 00:08:07,010
I could re-write it as -- let's see, what could I do?

159
00:08:07,009 --> 00:08:08,819
Let me take the x out of that side.

160
00:08:08,819 --> 00:08:18,449
So I could have y squared over 2 minus x squared over 2 is

161
00:08:18,449 --> 00:08:21,699
equal to -- and let me subtract this from that side so I get

162
00:08:21,699 --> 00:08:24,069
the constants all on the right hand side of the equation.

163
00:08:24,069 --> 00:08:26,839
c2 minus c1.

164
00:08:26,839 --> 00:08:29,959
I just took a c1, put it on the right hand side, took the x

165
00:08:29,959 --> 00:08:31,529
squared over 2, put it on the left hand side, that's

166
00:08:31,529 --> 00:08:32,429
why it's negative.

167
00:08:32,429 --> 00:08:33,500
And we didn't know.

168
00:08:33,500 --> 00:08:34,990
We said this could be any constant and this

169
00:08:34,990 --> 00:08:35,759
could be any constant.

170
00:08:35,759 --> 00:08:39,200
So the difference between two arbitrary constants could just

171
00:08:39,200 --> 00:08:40,610
be a third arbitrary constant.

172
00:08:40,610 --> 00:08:43,720
So I'll just re-write that as c.

173
00:08:43,720 --> 00:08:48,620
So the general solution to this differential equation is y

174
00:08:48,620 --> 00:08:56,080
squared over 2 minus x squared over 2 is equal to c.

175
00:08:56,080 --> 00:08:57,810
Actually let's do something else just to clean it up a

176
00:08:57,809 --> 00:09:00,089
little bit, because once again this could be any constant.

177
00:09:00,090 --> 00:09:02,870
So let's multiply both sides of this equation by 2 and

178
00:09:02,870 --> 00:09:08,740
you get y squared minus x squared equal to 2c.

179
00:09:08,740 --> 00:09:11,279
Well now this is still any constant number, so we could

180
00:09:11,279 --> 00:09:12,459
still write this as a c.

181
00:09:12,460 --> 00:09:17,070
So we have y squared minus x squared is equal to c.

182
00:09:17,070 --> 00:09:22,570
Now let's use our initial conditions to see what c is.

183
00:09:22,570 --> 00:09:29,110
So when y is 1, so 1 squared minus when x is 1, 1 squared is

184
00:09:29,110 --> 00:09:34,769
equal to c, this is 1 minus 1, so it's 0, right?

185
00:09:34,769 --> 00:09:36,590
c is equal to 0.

186
00:09:36,590 --> 00:09:39,570
So what is the particular solution to this

187
00:09:39,570 --> 00:09:40,350
differential equation?

188
00:09:40,350 --> 00:09:42,480
I'll do it in green.

189
00:09:42,480 --> 00:09:48,350
It is y squared minus x squared is equal to 0.

190
00:09:48,350 --> 00:09:50,269
Or we could add x squared to both sides of that.

191
00:09:50,269 --> 00:09:55,509
We could also write it as y squared is equal to x squared.

192
00:09:55,509 --> 00:09:58,870
Now you might be tempted to take the square root of both

193
00:09:58,870 --> 00:10:02,490
sides of this and say that y is equal to x.

194
00:10:02,490 --> 00:10:07,370
The reason why this would not be accurate is because here x

195
00:10:07,370 --> 00:10:14,289
could be minus 2 and y could be plus 2 or vice versa.

196
00:10:14,289 --> 00:10:16,559
So this would satisfy this equation, but it would not

197
00:10:16,559 --> 00:10:17,709
satisfy this equation.

198
00:10:17,710 --> 00:10:20,379
So be careful when you take that square root.

199
00:10:20,379 --> 00:10:22,179
You have to worry about the plus or minus.

200
00:10:22,179 --> 00:10:23,019
Let's do one more.

201
00:10:23,019 --> 00:10:29,750


202
00:10:29,750 --> 00:10:36,210
So this one says the derivative of y with respect to x is equal

203
00:10:36,210 --> 00:10:42,129
to -- OK, so this is even a little bit more interesting.

204
00:10:42,129 --> 00:10:47,774
This is equal to the square root of x over y.

205
00:10:47,774 --> 00:10:51,199


206
00:10:51,200 --> 00:10:56,890
And it says that y is equal to 4 when is equal to 1.

207
00:10:56,889 --> 00:10:58,220
Yup, I think I'm reading that right.

208
00:10:58,220 --> 00:11:00,670
We could do this one very similarly.

209
00:11:00,669 --> 00:11:03,959
So let's just do the same step.

210
00:11:03,960 --> 00:11:06,660
So multiply both sides of the equation times dx.

211
00:11:06,659 --> 00:11:11,939
So you get dy is equal to -- and just to skip a step, I'm

212
00:11:11,940 --> 00:11:15,630
going to re-write square root of x over y is the square root

213
00:11:15,629 --> 00:11:18,210
of x over the square root of y, and I multiply both

214
00:11:18,210 --> 00:11:21,180
sides of that times dx.

215
00:11:21,179 --> 00:11:23,009
Now let's multiply both sides of this equation

216
00:11:23,009 --> 00:11:25,799
by the square root of y.

217
00:11:25,799 --> 00:11:29,529
And I'm just going to re-write it as y to the 1/2 power.

218
00:11:29,529 --> 00:11:31,309
That's the same thing as the square root of y.

219
00:11:31,309 --> 00:11:35,729
dy is equal to x to the 1/2 power dx.

220
00:11:35,730 --> 00:11:37,399
I just multiplied that there and re-wrote

221
00:11:37,399 --> 00:11:39,309
it as y to the 1/2.

222
00:11:39,309 --> 00:11:45,319
So what is the antiderivative of y to the 1/2 power?

223
00:11:45,320 --> 00:11:54,780
Well it's just this plus 1, so it's y the 3/2 power, and then

224
00:11:54,779 --> 00:11:56,959
times the inverse of this.

225
00:11:56,960 --> 00:11:57,940
So times 2/3.

226
00:11:57,940 --> 00:12:00,730


227
00:12:00,730 --> 00:12:03,320
Or you could have said divided by this, either way.

228
00:12:03,320 --> 00:12:05,610
And if you're not sure about this, because sometimes it is

229
00:12:05,610 --> 00:12:08,560
confusing with the fractions and whatever else,

230
00:12:08,559 --> 00:12:09,669
take the derivative.

231
00:12:09,669 --> 00:12:12,539
3/2 times 2/3 is 1, and then you subtract this 1 from

232
00:12:12,539 --> 00:12:14,219
the exponent and you get y to the 1/2.

233
00:12:14,220 --> 00:12:15,139
So that works.

234
00:12:15,139 --> 00:12:19,539
And of course, we kind of go through the same drill, plus c1

235
00:12:19,539 --> 00:12:21,709
is equal to -- it's going to be the same thing on this side

236
00:12:21,710 --> 00:12:29,460
-- 2/3 x to the 3/2 plus c2, and what can we do?

237
00:12:29,460 --> 00:12:32,889
Well let's take the x to the left hand side of this

238
00:12:32,889 --> 00:12:42,990
equation, so we get 2/3 y to the 3/2 minus 2/3 x to the 3/2

239
00:12:42,990 --> 00:12:49,190
is equal to c2 minus c1, which we can just re-write as c.

240
00:12:49,190 --> 00:12:52,820
And let's multiply both sides of this equation by 3/2, so

241
00:12:52,820 --> 00:12:54,440
these two will become 1.

242
00:12:54,440 --> 00:13:03,980
So you get y to 3/2 minus x to the 3/2 is equal to -- well

243
00:13:03,980 --> 00:13:05,970
what's 3/2 times some constant c.

244
00:13:05,970 --> 00:13:07,800
But we didn't even know what it was, we haven't solved for

245
00:13:07,799 --> 00:13:08,799
it, so we can still write c.

246
00:13:08,799 --> 00:13:10,059
I hope that doesn't confuse you.

247
00:13:10,059 --> 00:13:12,099
3/2 c, we didn't know what it was.

248
00:13:12,100 --> 00:13:14,480
We could call this c3 and now this is c4.

249
00:13:14,480 --> 00:13:17,440
It's a different constant, but we still have to solve for it.

250
00:13:17,440 --> 00:13:19,790
And now let's use our initial conditions.

251
00:13:19,789 --> 00:13:24,724
So the initial conditions tell us that 4 -- and they don't

252
00:13:24,725 --> 00:13:25,639
have to be initial conditions.

253
00:13:25,639 --> 00:13:29,139
You can kind of say conditions or points where we know that

254
00:13:29,139 --> 00:13:33,830
the particular solution to this differential equation

255
00:13:33,830 --> 00:13:35,370
are satisfied.

256
00:13:35,370 --> 00:13:45,669
So 4 to the 3/2 minus 1 to the 3/2 is equal to c.

257
00:13:45,669 --> 00:13:46,829
What's 4 to the 3/2?

258
00:13:46,830 --> 00:13:51,620
So 4 to the 1/2 is 2, and then that to the 3 power, that's 8.

259
00:13:51,620 --> 00:13:54,129
Then 1 to any power, but especially in this case, 1 to

260
00:13:54,129 --> 00:13:56,439
the square root of 1 is 1.

261
00:13:56,440 --> 00:13:58,440
Then to the third power is 1.

262
00:13:58,440 --> 00:14:02,890
So 8 minus 1 is equal to c.

263
00:14:02,889 --> 00:14:05,639
And so c is equal to 7.

264
00:14:05,639 --> 00:14:09,629
So the particular solution of this differential equation I

265
00:14:09,629 --> 00:14:19,966
will do in a different color, and it is y to the 3/2 minus

266
00:14:19,966 --> 00:14:24,620
x to the 3/2 is equal to 7.

267
00:14:24,620 --> 00:14:25,000