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Understanding Differential Equations: Solutions and Graphical Representations

Learn about differential equations, general and particular solutions, graphical representations, and how to solve them, with examples and practice problems. Explore separable differential equations and solve for various functions.

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Understanding Differential Equations: Solutions and Graphical Representations

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  1. 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) 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? MAT 146

  3. What is a Solution to a Differential Equation? • A general solutionto a differential equation is a family of functions that satisfies a given differential equation. • A particular solutionto 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. MAT 146

  4. DE Warm-Ups • For the differential equation here, what are the constant solutions? • Solve this initial-value problem: MAT 146

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  12. 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 MAT 146

  13. y’ = 2x – y with initial conditions (0,0) MAT 146

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  21. Solving Differential Equations Solve for y: y’ = −y2 MAT 146

  22. Separable Differential Equations MAT 146

  23. Separable Differential Equations Solve for y: y’ = 3xy MAT 146

  24. Separable Differential Equations Solve for z: dz/dx+ 5ex+z = 0 MAT 146

  25. Separable Differential Equations MAT 146

  26. Separable Differential Equations MAT 146

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