Calculus II (MAT 146) Dr. Day Wednesday, March 7, 2018

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Presentation transcript:

Calculus II (MAT 146) Dr. Day Wednesday, March 7, 2018 Differential Equations (Chapter 9) What is a Differential Equation? (9.1) What is a Solution to a Differential Equation? (9.1) Graphical Representations of Solutions to Differential Equations (9.2) Wednesday, March 7, 2018 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, March 7, 2018 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, March 7, 2018 MAT 146

DE Questions (1) For the differential equation y’ + 2y = 2ex, Leonard claims that the following function is a solution. Describe and illustrate at least two different ways we can verify or refute Leonard’s claim. (2) Repeat problem (1) for the following differential equation and the proposed solution. Wednesday, March 7, 2018 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, March 7, 2018 MAT 146

DE Questions (4) Solve this initial-value problem: Wednesday, March 7, 2018 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, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146

Wednesday, March 7, 2018 MAT 146