Separable Equations & Modeling


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Lecture Notes

Ordinary Differential Equations

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Section 1, Page 1 to page 2

Definition, including examples of order 0, 1, 2, and k. Homogeneous and inhomogeneous differential equations are defined.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Derivatives (section 2 of lecture 1)
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Separable Differential Equations

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Section 2, Page 2 to page 4

Step-by-step solutions to separable differential equations and initial value problems.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Differentials and Antidifferentiation (lecture 13), Ordinary Differential Equations (section 1 of this lecture)
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Applications of Differential Equations

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Section 3, Page 4 to page 6

Exponential growth as a differential equation. Worked examples of population growth, radioactive decay, and Newton's Law of Cooling.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Differential Equations (sections 1 and 2 of this lecture)
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Online Textbook Chapter

Introduction to Differential Equations

Document Document

Definition, including the order of a differential equation as well as linear, homogeneous, inhomogeneous, and separable differential equations.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None
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Practice Problems

Separable Differential Equations

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Problem 1 (page 1 to page 2)

Three part question which involves setting up and solving separable differential equations.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Pages 3 to 4

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Exam Questions

Separable Ordinary Differential Equation

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Problem 3 (page 5)

Solving a separable ordinary differential equation with a given initial condition.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Page 3

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Differential Equations

Document PDF
Problem 10 (page 1) to problem 11 (page 1)

Two questions, one of which involves solving a first order differential equation and the other of which involves setting up and solving a differential equation for the temperature of a fish being cooked.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 2

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Differential Equations

Document PDF
Problem 5b (page 1)

Finding the solution to a first order differential equation.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Differential Equation

Document PDF
Problem 5b (page 2)

Finding the solution to a first order differential equation.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 2

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Differential Equations

Document PDF
Problem 7 (page 1)

Using separation of variables to find the solutions to a differential equation and describing the graphs of these solutions.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)#

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Differential Equations: Separation of Variables

Document PDF - 2.2 MB
Problem 3F-1 (page 25) to problem 3F-8 (page 26)

Eight questions which involve solving separable differential equations, including questions about Newton's Law of Cooling and about air pressure at different altitudes.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF - 4.0 MB) Pages 44 to 47

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Java Applet

First Order ODE

Java Applet Java Applet
Requires Java Virtual Machine

Applet for plotting the solution to a specified differential equation of one variable with a specified initial condition, along with the approximations given by the left hand, trapezoid, and Runge-Kutta rules.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differential Equations (OT26.1)
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Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound