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Unformatted text preview: Differential Equations&Mathematica 6 Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©20012007 4 Publisher: Math Everywhere, Inc.
Version 6.0 2 DE.01 Transition from Calculus to DiffEq:
The Exponential Differential Equation
y¢ @tD + r [email protected] = f @tD LITERACY What you need to know when you're away from the machine. · L.1)
Here are three diffeqs:
a) y£ @tD + 0.2 [email protected] = 0
b) y£ @tD  0.2 [email protected] = 0
c) y£ @tD + 0.1 [email protected] = 0.
Which of these diffeqs is solved by
[email protected] = 13 E0.2 t ?
· L.2)
Here are three diffeqs:
a) y£ @tD + [email protected] = E2 t
b) y£ @tD  [email protected] = E2 t
c) y£ @tD = E2 t .
Which of these diffeqs is solved by
[email protected] = 4 Et + E2 t ?
· L.3)
Here are three diffeqs:
a) y£ @tD + 2 t [email protected] = 0
b) y£ @tD  2 t [email protected] = 0
c) y£ @tD + [email protected] = 0.
Which of these diffeqs is solved by
2
[email protected] = 8 Et ? · L.4)
Here are three diffeqs:
a) y'' @tD + 4 [email protected] = 0
b) y'' @tD + 9 [email protected] = 0
c) y'' @tD + 16 [email protected] = 0.
Which of these diffeqs is solved by
[email protected] = 5 [email protected] tD ? · L.5)
Just to see that you can handle yourself away from the machine, come up with formulas for
the solutions of the following diffeqs:
a) y£ @tD + 0.3 [email protected] = 5 Et with [email protected] = 5.
b) y£ @tD  0.3 [email protected] = 5 with [email protected] = 1.
c) y£ @tD + [email protected] = [email protected]  3D with [email protected] = 2.
· L.6)
You know that y£ @tD = a [email protected] for all t's and you know that a is positive and that [email protected] is
positive. Does [email protected] go up or down as t advances from left to right? · L.7)
If a is negative , does the solution of y£ @tD = a [email protected] with [email protected] = 10 go up or down as t
advances from left to right. Can the solution ever go negative?
Why or why not?
· L.8)
The solution y[t] of
y£ @tD + r [email protected] = f @tD with [email protected] = starter
is given by
t
[email protected] = Er t starter + Er t Ÿ0 Er s f @sD „ s.
Explain this:
If r > 0, then, no matter what starter is, the plot of y[t] will eventually merge with the plot
of
t
[email protected] = Er t Ÿ0 Er s f @sD „ s.
· L.9)
Here are three plots of solutions of a certain forced exponential diffeq: 5 10 15 20 t 2
4
6
You are given that all three plots are all either solutions of
a) y£ @tD + 0.6 [email protected] = 4 [email protected] tD + [email protected] tD
or solutions of
b) y£ @tD  0.6 [email protected] = 4 [email protected] tD + [email protected] tD.
Make your choice and say how you arrived at it.
· L.10)
The solution y[t] of
y£ @tD + r [email protected] = f @tD with [email protected] = starter
is given by
t
[email protected] = Er t starter + Er t Ÿ0 Er s f @sD „ s.
Explain this:
If r < 0 then, unless starter = 0, the plot of y[t] will NOT eventually merge with the plot of
t
[email protected] = Er t Ÿ0 Er s f @sD „ s.
· L.11)
Here are three plots of solutions of a certain forced exponential diffeq: 150
100
50
5 10 15 20 t  50
You are given that all three plots are all either solutions of
a) y£ @tD + 0.15 [email protected] = [email protected] tD + 2 [email protected] tD
or solutions of
b) y£ @tD  0.15 [email protected] = [email protected] tD + 2 [email protected] tD.
Make your choice and say how you arrived at it.
· L.12)
Here are plots of the solutions of
diffeq a): y' @tD + 0.01 [email protected] = 0
diffeq b): y£ @tD + 0.3 [email protected] = 0,
diffeq c): y£ @tD + 1.3 [email protected] = 0,
diffeq d): y' @tD + 2.3 [email protected] = 0,
all with the same starter [email protected] = 7.
The plots are not in order. Your job is to match the differential equation with the plot of
its solution. Plot A Plot B 7
6
5
4
3
2
1
t
0 2 4 6 8 1012 7
6
5
4
3
2
1
t
0 2 4 6 8 1012 Plot C
7
6
5
4
3
2
1
t
0 2 4 6 8 1012
diffeq a) > Plot.......
diffeq c) > Plot....... Plot D
7
6
5
4
3
2
1
t
0 2 4 6 8 1012
diffeq b) > Plot.......
diffeq d) > Plot....... diffeq a) > Plot.......
diffeq c) > Plot....... diffeq b) > Plot.......
diffeq d) > Plot....... · L.13)
Do you expect solutions of
y£ @tD  0.6 [email protected] = 0
to decay to 0 as t gets large?
Why or why not?
· L.14)
Here are plots of solutions of the forced exponential diffeq
y£ @tD + 0.8 [email protected] = f @tD
with
[email protected] = 4.0
for four choices of forcing functions f[t]: Plot A 5 2
4 70
60
50
40
30 Plot B 6
4
2 10 15 20 t 6
4
2 20
10 5 2
4 Plot C 10 15 20 t Plot D 6
4
2 · L.18)
Here are plots of solutions of
y£ @tD = 0.6 [email protected] with [email protected] = 2
and
[email protected]
y£ @tD = 0.6 [email protected] I1  100 M with [email protected] = 2 : 6
4
2 t
t
5 10 15 20
5 10 15 20
2
2
4
4
The four forcing functions f[t] used in these plots are:
[email protected] = 0.2 t,
:
[email protected] = 5 [email protected]  6D,
0.1 t
[email protected] = 2.3 E
[email protected],
[email protected] = 5 [email protected] tD.
Your job is to match the plots to the forcing functions:
f1[t]> Plot..... f2[t]> Plot.....
f3[t]> Plot..... f4[t]> Plot.....
· L.15)
Give the formula for the solution of
y£ @xD = 0.2 [email protected] with [email protected] = 1
and pencil in a rough sketch of the solution on the axes below. t
1
2
3
4
5
6
Explain why as t advances from 0, the plots of both of these solutions had no choice but to
share a lot of ink initially.
Explain why for larger t's, the plots had no choice but to pull apart, with one plot
eventually sailing way above the other.
· L.19)
Here are six plots: They are plots of:
a. a solution of y£ @xD = r [email protected] with r > 0.
b. a solution of y£ @xD = r [email protected] with r < 0.
c. none of the above
Which is which? The plotted point is on the curve. [email protected]
60
50
40
30
20 Plot 1 y
14
12
10 0 2 4 6 810
12 Plot 2 x [email protected]
6
4
2
x
 2 10 20 30 40 Plot 3 8
6
4
2 [email protected]
8
6
4 Plot 5 2 4 6 8 x 10 12 · L.16)
Give the formula for the solution of
y£ @xD =  0.5 [email protected] with [email protected] = 6
and pencil in a rough sketch of the solution on the axes below. The plotted point is on the curve. y
8 [email protected]
6000
5000
4000
3000
2000 5 10 15 20 [email protected]
8
6
4 Plot 4 10 20 30 40 [email protected]
12
10
8
6
4 4 2 2 4 6 8 x · L.17)
You are given that
[email protected] = 37.3 E0.045 t
Write down a differential equation that [email protected] solves.
Include starter data. x Plot 6 x
x
10 20 30 40
10 30 50
20 40 60
Plot 1 >............ Plot 2 >............ Plot 3 >............
Plot 4 >............ Plot 5 >............ Plot 6 >............ · L.20)
aL You are given a solution [email protected] of
y£ @tD + 5 [email protected] = 6 [email protected]  4.7D
What happens to [email protected] at t = 4.7 ? bL How does your response indicate that
[email protected]  4.7D ! 2 [email protected]  4.7D ? 6 x · L.21)
Here is the part of the plot of the solution of
y£ @tD + 1.1 [email protected] = 5 [email protected]  3D with [email protected] = 5.
Your job is to sketch in the rest of the plot. [email protected]
5
4
3
2
1
0
0
2
· L.22)
a) Write down the values of
Ÿ0 4.99999 Ÿ0 5.00001 4 6 8 t [email protected]  5D „ t
[email protected]  5D „ t Ÿ5.00001 [email protected]  5D „ t.
100 Et [email protected]  3D „ t b) Write down the values of
Ÿ0 2.99999 3.000001
Ÿ2.999999
100 000
Ÿ3.00001 Et [email protected]  3D „ t Et [email protected]  3D „ t c) Here's a plot of UnitStep[t  2]:
1.0
0.8
0.6
0.4
0.2 1
2
3
And here's a plot of UnitStep[t  4]: 4 5 6 t 1.0
0.8
0.6
0.4
0.2
t
1
2
3
4
5
6
Digest the two plots and explain the result of this Mathematica calculation:
[email protected], xD; AssumingBt œ Reals, ‡ Sin@xD DiracDelta@x  5D „ xF
t HeavisideTheta@ 5 + tD Sin@5D
0 ...
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This note was uploaded on 09/21/2011 for the course MATH 285 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff
 Differential Equations, Equations

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