Calculus II (MAT 146) Dr. Day Monday, March 5, 2018

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Presentation transcript:

Calculus II (MAT 146) Dr. Day Monday, March 5, 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) Monday, March 5, 2018 MAT 146

(A) What Does it Mean for Something to be a Solution to an Equation? (i) Is x = −1/5 a solution to the equation 4x + 7 = 2x – 3 ? (ii) State all solutions to the equation x2 – x = 12. (iii) Is x = 4 a solution to the equation 3ex = 12 ? (B) What Information About a Function is Revealed Through Its First Derivative? g’(x) = –8 –6x2. Gordon looked at the derivative of g(x) and shouted, “The function g is always decreasing!” How did he know? Monday, March 5, 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? Monday, March 5, 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. Monday, March 5, 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. Monday, March 5, 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? Monday, March 5, 2018 MAT 146

DE Questions (4) Solve this initial-value problem: Monday, March 5, 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. Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 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 Monday, October 16, 2017 MAT 146

with initial conditions (0,0) y’ = 2x – y with initial conditions (0,0) Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Monday, October 16, 2017 MAT 146

Solving Differential Equations Solve for y: y’ = −y2 Monday, October 16, 2017 MAT 146

Separable Differential Equations Monday, October 16, 2017 MAT 146

Separable Differential Equations Solve for y: y’ = 3xy Monday, October 16, 2017 MAT 146

Separable Differential Equations Solve for z: dz/dx+ 5ex+z = 0 Monday, October 16, 2017 MAT 146

Separable Differential Equations Monday, October 16, 2017 MAT 146

Separable Differential Equations Monday, October 16, 2017 MAT 146