1. Calculus II (MAT 146) Dr. Day Monday, Oct 16, 2017
Differential Equations (Chapter 9)
What is a differential equation? (9.1)
Solutions to differential equation (9.1)
Monday, October 16, 2017
MAT 146

2. (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, October 16, 2017
MAT 146

3. 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, October 16, 2017
MAT 146

4. 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, October 16, 2017
MAT 146

5. 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, October 16, 2017
MAT 146

6. 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, October 16, 2017
MAT 146

7. DE Questions (4) Solve this initial-value problem:
Monday, October 16, 2017
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8. 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.
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16. 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
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17. with initial conditions (0,0)
y’ = 2x – y
with initial conditions (0,0)
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25. Solving Differential Equations
Solve for y: y’ = −y2
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26. Separable Differential Equations
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27. Separable Differential Equations
Solve for y: y’ = 3xy
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28. Separable Differential Equations
Solve for z: dz/dx+ 5ex+z = 0
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29. Separable Differential Equations
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30. Separable Differential Equations
Monday, October 16, 2017
MAT 146