Today. Introduction to Differential Equations. Linear DE ( y = ky ) Nonlinear DE (e.g. y = y (1-y) ) Qualitative analysis (phase line)
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1 Today Introduction to Differential Equations Linear DE ( y = ky ) Nonlinear DE (e.g. y = y (1-y) ) Qualitative analysis (phase line)
2 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. (A) C (t) = k C(t) where k>0. (B) C (t) = k C(t) where k<0. (C) C(t) = C 0 e kt. (D) C (t) = C 0 e -kt.
3 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. (A) C (t) = k C(t) where k>0. (B) C (t) = k C(t) where k<0. (C) C(t) = C 0 e kt. (D) C (t) = C 0 e -kt.
4 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. (A) C (t) = k C(t) where k>0. <---solution grows! (B) C (t) = k C(t) where k<0. (C) C(t) = C 0 e kt. (D) C (t) = C 0 e -kt.
5 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. (A) C (t) = k C(t) where k>0. <---solution grows! (B) C (t) = k C(t) where k<0. (C) C(t) = C 0 e kt. (D) C (t) = C 0 e -kt. <---if k<0, this might be the solution but it s not a DE.
6 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. (A) C (t) = k C(t) where k>0. <---solution grows! (B) C (t) = k C(t) where k<0. (C) C(t) = C 0 e kt. (D) C (t) = C 0 e -kt. <---if k<0, this might be the solution but it s not a DE. <---this is not a DE either.
7 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. C (t) = -r C(t) where r > 0. Solution: (A) C(t) = e -rc (B) C(t) = 17e -rt (C) C(t) = -rc 2 /2 (D) C(t) = 5e rt
8 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. C (t) = -r C(t) where r > 0. Solution: (A) C(t) = e -rc (B) C(t) = 17e -rt (C) C(t) = -rc 2 /2 (D) C(t) = 5e rt
9 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. C (t) = -r C(t) where r > 0. Solution: (A) C(t) = e -rc In fact, C(t) = C 0 e -rt is a solution for all values of C 0 - show on board. (B) C(t) = 17e -rt (C) C(t) = -rc 2 /2 (D) C(t) = 5e rt
10 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. C (t) = -r C(t) where r > 0. Solution: (A) C(t) = e -rc (B) C(t) = 17e -rt (C) C(t) = -rc 2 /2 (D) C(t) = 5e rt In fact, C(t) = C 0 e -rt is a solution for all values of C 0 - show on board. DEs are often given with an initial condition (IC) e.g. C(0)=17 which can be used to determine C 0.
11 Differential equations (DE) Carbon dating: The amount of Carbon-14 in a sample decreases at a rate proportional to how much C-14 is present. Write down a DE for the amount of C-14 at time t. C (t) = -r C(t) where r > 0. Solution: (A) C(t) = e -rc (B) C(t) = 17e -rt (C) C(t) = -rc 2 /2 (D) C(t) = 5e rt In fact, C(t) = C 0 e -rt is a solution for all values of C 0 - show on board. DEs are often given with an initial condition (IC) e.g. C(0)=17 which can be used to determine C 0. DE + IC is called an Initial Value Problem (IVP)
12 Summary of what you should be able to do
13 Summary of what you should be able to do Take a word problem of the form Quantity blah changes at a rate proportional to how much blah there is and write down the DE:
14 Summary of what you should be able to do Take a word problem of the form Quantity blah changes at a rate proportional to how much blah there is and write down the DE: Q (t) = k Q(t)
15 Summary of what you should be able to do Take a word problem of the form Quantity blah changes at a rate proportional to how much blah there is and write down the DE: Q (t) = k Q(t) Write down the solution to this equation:
16 Summary of what you should be able to do Take a word problem of the form Quantity blah changes at a rate proportional to how much blah there is and write down the DE: Q (t) = k Q(t) Write down the solution to this equation: Q(t) = Q 0 e kt
17 Summary of what you should be able to do Take a word problem of the form Quantity blah changes at a rate proportional to how much blah there is and write down the DE: Q (t) = k Q(t) Write down the solution to this equation: Q(t) = Q 0 e kt Determine k and Q 0 from given values or %ages of Q at two different times (i.e. data).
18 Summary of what you should be able to do Take a word problem of the form Quantity blah changes at a rate proportional to how much blah there is and write down the DE: Q (t) = k Q(t) Write down the solution to this equation: Q(t) = Q 0 e kt Determine k and Q 0 from given values or %ages of Q at two different times (i.e. data). Determine half-life/doubling time from data or k.
19 DEs - a broad view
20 DEs - a broad view We have talked about linear DEs so far: y = ky
21 DEs - a broad view We have talked about linear DEs so far: y = ky A linear DE is one in which the y and the y appear linearly (more later about y = a + ky).
22 DEs - a broad view We have talked about linear DEs so far: y = ky A linear DE is one in which the y and the y appear linearly (more later about y = a + ky). Some nonlinear equations: v = g-v 2, y = -sin(y), (h ) 2 = bh.
23 DEs - a broad view We have talked about linear DEs so far: y = ky A linear DE is one in which the y and the y appear linearly (more later about y = a + ky). Some nonlinear equations: v = g-v 2, y = -sin(y), (h ) 2 = bh. object falling through air pendulum under water water draining from a vessel
24 DEs - a broad view
25 DEs - a broad view Where do nonlinear equations come from?
26 DEs - a broad view Where do nonlinear equations come from? Population growth: N = bn-dn = kn (linear) where b is per-capita birth rate, d is percapita death rate and k=b-d.
27 DEs - a broad view Where do nonlinear equations come from? Population growth: N = bn-dn = kn (linear) where b is per-capita birth rate, d is percapita death rate and k=b-d. Suppose the per-capita death rate is not constant but increases with population size (more death at high density) so d = cn.
28 DEs - a broad view Where do nonlinear equations come from? Population growth: N = bn-dn = kn (linear) where b is per-capita birth rate, d is percapita death rate and k=b-d. Suppose the per-capita death rate is not constant but increases with population size (more death at high density) so d = cn. N = bn-(cn)n = bn-cn 2
29 DEs - a broad view
30 DEs - a broad view dn dt = bn cn 2
31 DEs - a broad view dn dt = bn cn 2 This is called the logistic equation, usually written as dn dt = rn N 1. K
32 DEs - a broad view dn dt = bn cn 2 This is called the logistic equation, usually written as dn dt = rn N 1. K where r=b and K=1/c. This is a nonlinear DE because of the N 2.
33 Qualitative analysis
34 Qualitative analysis Finding a formula for a solution to a DE is ideal but what if you can t?
35 Qualitative analysis Finding a formula for a solution to a DE is ideal but what if you can t? Qualitative analysis - extract information about the general solution without solving.
36 Qualitative analysis Finding a formula for a solution to a DE is ideal but what if you can t? Qualitative analysis - extract information about the general solution without solving. Steady states Slope fields Stability of steady states Plotting y versus y (state space/phase line)
37 x 0 = x(1 x) velocity position Steady state. Where can you stand so that the DE tells you not to move? (A) x=-1 (B) x=0 (C) x=1/2 (D) x=1 This is the logistic eq with r=1, K=1.
38 x 0 = x(1 x) velocity position Steady state. Where can you stand so that the DE tells you not to move? (A) x=-1 (B) x=0 (C) x=1/2 (D) x=1 This is the logistic eq with r=1, K=1.
39 x 0 = x(1 x) velocity position Steady state. Where can you stand so that the DE x(t) tells you not to move? (A) x=-1 1 (B) x=0 (C) x=1/2 (D) x=1 0 This is the logistic eq with r=1, K=1. t A steady state is a constant solution.
40 y = -y(y-1)(y+1) What are the steady states of this equation?
41 x 0 = x(1 x) velocity position Slope field x(t) 1 0 t
42 x 0 = x(1 x) velocity position Slope field Slope field. x(t) 1 0 t
43 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 0 t
44 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values When x(t)=1/2 what is x? (A) 0 (B) 1/4 (C) 1/2 (D) 1 1 1/2 0 t
45 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values When x(t)=1/2 what is x? (A) 0 (B) 1/4 (C) 1/2 (D) 1 1 1/2 0 t
46 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
47 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
48 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
49 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
50 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
51 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
52 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
53 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values 1 1/2 0 t
54 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. 1 1/2 0 t
55 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. 1 1/2 0 t
56 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. 1 1/2 0 t
57 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. 1 1/2 0 t
58 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. 1 1/2 0 t
59 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. 1 1/2 0 t
60 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
61 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
62 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
63 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
64 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
65 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
66 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
67 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
68 x 0 = x(1 x) velocity position Slope field Slope field. x(t) At any t, don t know x yet so plot all possible x values Now draw them for all t. Solution curves must be tangent to slope field everywhere. 1 1/2 0 t
69 y = -y(y-1)(y+1) What are the steady states of this equation? Draw the slope field for this equation. Include the steepest slope element in each interval between steady states and two others (roughly).
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