Unit 7 Differential Equations

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ENDURING UNDERSTANDING
FUN-7 Solving differential equations allows us to determine functions and develop models.

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Topic Name

Essential Knowledge

7.1 Modeling Situation with Differential Equations

LEARNING OBJECTIVE

FUN-7.A Interpret verbal statements of problems as differential equations involving a derivative expression.

FUN-7.A.1 Differential equations relate a function of an independent variable and the function’s derivatives.

Blog Posts

Differential Equations The basics and definitions.

Good Question 2: 2002 BC 5 A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions  Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

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7.2 Verifying Solutions for Differential Equations

LEARNING OBJECTIVE

FUN-7.B Verify solutions to differential equations.

FFUN-7.B.1 Derivatives can be used to verify that a function is a solution to a given differential equation.

 

FUN-7.B.2 There may be infinitely many general solutions to a differential equation.

Blog Posts

An Exploration in Differential Equations  An exploration covering pretty much all of the ideas in differential equation based on 2018 AB 6.  The exploration is here and the solutions here..

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7.3 Sketching Slope Fields

LEARNING OBJECTIVE

FUN-7.C Estimate solutions to differential equations.

FUN-7.C.1 A slope field is a graphical representation of a differential equation on a finite set of points in the plane.
FUN-7.C.2 Slope fields provide information about the behavior of solutions to first-order differential equations.

Blog Posts

Slope Fields Graphical solutions: The solution is lurking in the slope field.

Good Question 2: 2002 BC 5 A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions  Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

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7.4 Reasoning Using Slope Fields

LEARNING OBJECTIVE

FUN-7.C Estimate solutions to differential equations.

FUN-7.C.3 Solutions to differential equations are functions or families of functions.

Blog Posts

Slope Fields Graphical solutions: The solution is lurking in the slope field.

Good Question 2: 2002 BC 5 A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

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7.5 Approximating Solutions Using Euler’s Method BC ONLY

LEARNING OBJECTIVE

FUN-7.C Estimate solutions to differential equations.

FUN-7.C.4 Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve. BC ONLY

Blog Posts

Euler’s Method (1-12-2015) Numerical solutions (BC only topic)

Euler’s Method for Making Money (2-25-2015) The connection between compound growth (compound interest) and Euler’s Method.

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7.6 Finding General Solutions Using Separation of Variables

LEARNING OBJECTIVE

FUN-7.D Determine general solutions to differential equations.

FUN-7.D.1 Some differential equations can be solved by separation of variables.
FUN-7.D.2 Antidifferentiation can be used to find general solutions to differential equations.

Blog Posts

Accumulation and Differential Equations  (2-1-2013) Solving differential equations without the “+C

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7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

LEARNING OBJECTIVE

FUN-7.E Determine particular solutions to differential equations.

FUN-7.E.1 A general solution may describe infinitely many solutions to a differential equation. There is only one particular solution passing through a given point.
FUN-7.E.2 The function F defined F\left( x \right)={{y}_{0}}+\int_{a}^{x}{{f\left( t \right)dt}} is a particular solution to the differential equation \frac{{dy}}{{dx}}=f\left( x \right) dy = fx ()dx , satisfying F\left( a \right)={{y}_{0}}
FUN-7.E.3 Solutions to differential equations may be subject to domain restrictions.

Blog Posts

Domain of a Differential Equation (4-7-2017) notes and examples on finding the domain of the solution of a differential equation

Accumulation and Differential Equations  (2-1-2013) Solving differential equations without the “+C

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7.8 Exponential Models with Differential Equations

LEARNING OBJECTIVES

FUN-7.F Interpret the meaning of a differential equation and its variables in context.

FUN-7.G Determine general and particular solutions for problems involving differential equations in context.

FUN-7.F.1 Specific applications of finding general and particular solutions to differential equations include motion along a line and exponential growth and decay.
FUN-7.F.2 The model for exponential growth and decay that arises from the statement “The rate of change of a quantity is proportional to the size of the quantity” is \frac{{dy}}{{dt}}=ky .
FUN-7.G.1 The exponential growth and decay model, \frac{{dy}}{{dt}}=ky , with initial condition y={{y}_{0}} when t = 0, has solutions of the form y={{y}_{0}}{{e}^{{kt}}} .

Blog Posts

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7.9 Logistic Models with Differential Equations BC ONLY

LEARNING OBJECTIVE

FUN-7.H Interpret the meaning of the logistic growth model in context. bc only

FUN-7.H.1 The model for logistic growth that arises from the statement “The rate of change of a quantity is jointly proportional to the size of the quantity and the difference between the quantity and the carrying capacity” is \frac{{dy}}{{dt}}=ky\left( {a-y} \right). BC ONLY
FUN-7.H.2 The logistic differential equation and initial conditions can be interpreted without solving the differential equation. BC ONLY
FUN-7.H.3 The limiting value (carrying capacity) of a logistic differential equation as the independent variable approaches infinity can be determined using the logistic growth model and initial conditions. BC ONLY
FUN-7.H.4 The value of the dependent variable in a logistic differential equation at the point when it is changing fastest can be determined using the logistic growth model and initial conditions. BC ONLY

Blog Posts

Logistics Growth – Real and Simulated

Type 6 Questions: Differential Equations  Summary and review notes.

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