dP kP L P dt P is the function L P largestmath121/Notes/Annotated_Online/... · 2017. 12. 15. ·...
Transcript of dP kP L P dt P is the function L P largestmath121/Notes/Annotated_Online/... · 2017. 12. 15. ·...
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Slope Fields - Logistic Growth - 2
dP
dt= k P (L− P )
For what values of P is the function k P (L− P ) largest?
(a) P = 0
(b) P = L
(c) P = 2L
(d) P =L
2
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Slope Fields - Logistic Growth - 3
Sketch the slope field associated with the differential equation
dP
dt= k P (L− P ).
On the slope field, draw several solutions using different initial conditions.
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Euler’s Method - 1
Euler’s Method
We can extend the idea of a slope field (a visual technique) to Euler’s method (anumerical technique). Euler’s method can be used to produce approximations ofthe curve y(x) that satisfy a particular differential equation. Here is the idea:
Knowing where you are in x and y, you look at the slope field at yourlocation, set off in that direction for a small distance, look again and adjustyour direction, set off in that direction for a small distance, etc.
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Euler’s Method - 2
Algorithmically,
• Start at a point (xi, yi)
• Compute the slope there, using the DE dydx
= f (xi, yi)
• Follow the slope for a step of ∆x:– xi+1 = xi + ∆x
– yi+1 = yi +dy
dx∆x︸ ︷︷ ︸
∆y
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Euler’s Method - 3
Follow this procedure for the differential equationdy
dx= x + y with initial con-
dition y(0) = 0.1. Use ∆x = 0.1.
x y slope ∆y = slope ·∆x
0 0.1 0.1 (0.1)(0.1) = 0.01
0.1 0.11
0.2
0.3
0.4
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Euler’s Method - 4
Here is a picture of the slope field fordy
dx= x + y. On this slope field, sketch
what you have done in creating the table of values. From the picture, wouldyou say the values for y(x) in your table are over-estimates or under-estimatesof the real y values?
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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Separation of Variables - 1
Separation of Variables
We have now considered both visual and approximate techniques for solving dif-ferential equations, which can be obtained with no calculus. The problem withthose approaches is that they do not result in formulas for the function y that wewant to identify.We (at last!) proceed to calculus-based techniques for finding a formula for y.
Consider the differential equationdy
dx= k y.
Treating dx and dy as separable units, transform the equation so that onlyterms with y are on the left, and only terms with x are on the right.
Place an integral sign in front of each side.
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Separation of Variables - 2
Evaluate the integrals.
Solve for y.
The solution gives a family of functions, one for each value of the integrationconstant. k is also a parameter, of course, but it is presumed to be specified in thedifferential equation.
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Separation of Variables - 3
As soon as we are given an initial value, say y(0) = 10, the solution becomesunique.Find the specific solution with initial value y(0) = 10.
If y0 > 0, this function describes exponential growth (k > 0) or decay (k < 0).
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Separation of Variables - 4
Use the method of separation of variables to solve the differential equation
dR
dx= 2R + 3,
and find the particular solution for which R(0) = 0.
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Classifying Differential Equations - 1
Classifying Differential EquationsFor any differential equation which is separable, we can at least attempt to find asolution using anti-derivatives. For equations which are not separable, we’ll needother techniques. It is important, as a result, to be able to tell the difference!Indicate which of the following differential equations are separable. For thosewhich are separable, set up the appropriate integrals to start solving for y.
• dydx
= x2
• dydx
=ey
x
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Classifying Differential Equations - 2
• dydx
= x + y
• dydx
= cos(x) cos(y)
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Classifying Differential Equations - 3
• dydx
= cos(xy) A. Separable B. Not separable
• dydx
= ex + ey A. Separable B. Not separable
• dydx
= e(x+y) A. Separable B. Not separable
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Classifying Differential Equations - 4
Note: all the original anti-derivatives we studied in first term are of the formdy
dx= f (x) and so y = F (x) =
∫f (x) dx.
E.g.dy
dx= x2
dy
dx= x cos(5x)
dy
dx=
x
1 + x2
These are all immediately separable.
The challenge is that most interesting scientific laws expressed in differential equa-tion form aren’t that easy to work with.