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

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.

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.

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

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

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

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.

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.

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).

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.

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

Classifying Differential Equations - 2

• dydx

= x + y

• dydx

= cos(x) cos(y)

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

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.