Calculus II (MAT 146) Dr. Day Wednesday, Oct 18, 2017

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Presentation transcript:

Calculus II (MAT 146) Dr. Day Wednesday, Oct 18, 2017 Differential Equations (Chapter 9) What is a differential equation? (9.1) Solutions to differential equation (9.1) Wednesday, October 18, 2017 MAT 146

What is a Differential Equation? A differential equation is an equation that contains one or more derivatives. Here’s a differential equation you have already solved: y’ = 2x What is the solution of this differential equation? Wednesday, October 18, 2017 MAT 146

What is a Solution to a Differential Equation? A general solution to a differential equation is a family of functions that satisfies a given differential equation. A particular solution to a differential equation (also called the solution to an initial-value problem) is a particular function that satisfies both a given differential equation and some specified ordered pair for the function. Wednesday, October 18, 2017 MAT 146

DE Questions (3) Population growth for a certain organism is modeled by this differential equation, with P measured in millions, t in years: (a) For what values of the population P will the population be growing? (b) For what values of the population P will the population be diminishing? (a) What, if any, are the equilibrium values of the population? Wednesday, October 18, 2017 MAT 146

DE Warm-Ups For the differential equation here, what are the constant solutions? For the following differential equation, determine whether any of the functions that follow are solutions. Wednesday, October 18, 2017 MAT 146

DE Questions (4) Solve this initial-value problem: Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

knowing that (0,0) satisfies y Solve the differential equation graphically by generating a slope field and then sketching in a solution: y’ = 2x – y knowing that (0,0) satisfies y Wednesday, October 18, 2017 MAT 146

with initial conditions (0,0) y’ = 2x – y with initial conditions (0,0) Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Wednesday, October 18, 2017 MAT 146

Solving Differential Equations Solve for y: y’ = −y2 Wednesday, October 18, 2017 MAT 146

Separable Differential Equations Wednesday, October 18, 2017 MAT 146

Separable Differential Equations Solve for y: y’ = 3xy Wednesday, October 18, 2017 MAT 146

Separable Differential Equations Solve for z: dz/dx+ 5ex+z = 0 Wednesday, October 18, 2017 MAT 146

Separable Differential Equations Wednesday, October 18, 2017 MAT 146

Separable Differential Equations Wednesday, October 18, 2017 MAT 146