Calculus 1120, Spring 2012 Dan Barbasch October 18 Dan Barbasch () - - PowerPoint PPT Presentation

Calculus 1120, Spring 2012

Dan Barbasch October 18

Dan Barbasch () Calculus 1120, Spring 2012 October 18 1 / 3

First Order Equations

y′ = f (x, y) A first order differential equation Initial Value Problem: Find a solution to a first order equation satisfying the extra condition y(x0) = y0.

Dan Barbasch () Calculus 1120, Spring 2012 October 18 2 / 3

First Order Equations

Motivation/Examples:

1 Exponential Growth/Decay, population growth/radioactive

decay/interest

2 Logistic Equation 3 Circuits 4 Newton’s Law of Cooling 5 Motion with resistance proportional to velocity 6 Mixture Problems Dan Barbasch () Calculus 1120, Spring 2012 October 18 2 / 3

First Order Equations

Solving Equations: Finding solutions directly, separable/linear equations Graphical Methods, slope fields Numerical Methods, Euler’s Method

Dan Barbasch () Calculus 1120, Spring 2012 October 18 2 / 3

Graphical Method

Finding an explicit solution is often not possible. There is a graphical way to get a sense of what the solutions look like. A solution curve y(x) to the equation y′ = f (x, y) has slope of the tangent line at any point (x, y) equal to f (x, y). Draw a small line at each (x, y) with slope f (x, y). This is called the slope field (direction field) of the equation. Choose a starting point, and sketch a curve whose tangent at each point is the slope field. These lines give you a qualitative sense of what the solutions look like.

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3

Graphical Method

dP dt = P(1 − P) When drawing a slope field, identify the regions where f (x, y) > 0, and where f (x, y) < 0. The solutions are increasing and decreasing respectively in these regions. At a point (x0, y0) where the function f (x, y) = 0 the slope of the tangent line is flat. In this example, the function f (t, P) = f (P) is independent of t. Equations y′ = g(y) are called autonomous. For such an equation, if g(y0) = 0, then y = y0 is a solution. It is called an equilibrium. In the example, P(t) = 0 and P(t) = 1 are

• equilibriums. P = 1 is stable, P = 0 is unstable.

A basic theorem about differential equations says that if f (x, y) is continuous in x and y, and continuously differentiable in y, then the solutions do not intersect. Same as saying that the equation has a unique solution with a given initial condition.

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3

Graphical Method

dP dt = P(1 − P)

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3

Graphical Method

dP dt = P(1 − P) Solution: P =

P0 P0+(1−P0)e−t = P0et 1+P0(et−1).

P0 = 2, P(t) = 2 2 − e−t , t > − ln 2! P0 = 1/2, P(t) = 1 2 1 1/2 + e−t/2 = 1 1 + e−t −∞ < t < ∞ P0 = −1, P(t) = −1 −1 + 2e−t = 1 1 − 2e−t t < ln 2! The last equation has a vertical asymptote t < ln 2! WARNING: There is NO long term behaviour for this solution. In general for P0 < 0, t < ln 1−P0

−P0

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3

Graphical Method

dy dx = y2 − x2.

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3

Graphical Method

dy dx = y2 − x2.

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3

Graphical Method

y′ = −y + 1 You can find a program that draws the slope field and solutions of an ODE y′ = f (x, y) at http://math.rice.edu/˜dfield/dfpp.html

Dan Barbasch () Calculus 1120, Spring 2012 October 18 3 / 3