Calculus II (MAT 146) Dr. Day Monday, March 19, 2018

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Presentation transcript:

Calculus II (MAT 146) Dr. Day Monday, March 19, 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 19, 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 19, 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 19, 2018 MAT 146

DE Warm-Ups For the differential equation here, what are the constant solutions? Solve this initial-value problem: Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 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, March 19, 2018 MAT 146

with initial conditions (0,0) y’ = 2x – y with initial conditions (0,0) Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Monday, March 19, 2018 MAT 146

Solving Differential Equations Solve for y: y’ = −y2 Monday, March 19, 2018 MAT 146

Separable Differential Equations Monday, March 19, 2018 MAT 146

Separable Differential Equations Solve for y: y’ = 3xy Monday, March 19, 2018 MAT 146

Separable Differential Equations Solve for z: dz/dx+ 5ex+z = 0 Monday, March 19, 2018 MAT 146

Separable Differential Equations Monday, March 19, 2018 MAT 146

Separable Differential Equations Monday, March 19, 2018 MAT 146