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

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1 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

2 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

3 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

4 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

5 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

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

7 Wednesday, October 18, 2017 MAT 146

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14 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

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

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22 Wednesday, October 18, 2017 MAT 146

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

24 Separable Differential Equations
Wednesday, October 18, 2017 MAT 146

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

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

27 Separable Differential Equations
Wednesday, October 18, 2017 MAT 146

28 Separable Differential Equations
Wednesday, October 18, 2017 MAT 146

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