Differential Equations

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Basic Concepts and Properties

Differential equations define a relationship between a function and its derivative. Begin with vectors and superposition to establish foundations in the basics before jumping into other themes.

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Linear Algebra and VectorsFree

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Differential Equations: BasicsFree

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First Order Differential Equations

Differential equations that only contain a first derivative are known as first order. Discover techniques to solve separable equations and apply to both linear and nonlinear examples. Also explore the concept of the slope field as a visual tool.

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Linear Equations and Integrating FactorsFree

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Linear equationsFree

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Integrating factorsFree

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Separable Equations

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Existence and Uniqueness

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Existence

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Uniqueness

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Exact Equations and Integrating Factors

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Exact equations

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Integrating factors

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Non-linear First Order Equations

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Slope Fields

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Second Order Differential Equations

Second order differential equations, those that contain a second derivative, are often difficult to solve. Explore both homogeneous and inhomogeneous equations, discover the Wronskian as a solution tool, and apply second order differential equations to forced oscillators.

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Homogeneous Equations with Constant Coefficients

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Wronskians

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Linear homogeneous equations

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The Wronskian

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Inhomogeneous Equations

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Forced Oscillators and Resonance

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Linear Systems

In many scenarios, one differential equation can interact with another. Learn about the basics of eigenvectors and eigenvalues in this theme and apply them to linear systems of differential equations.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

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Eigenvalues and Eigenvectors

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Eigenvalues

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Eigenvectors

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Theory of First Order Linear Systems

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Complex Eigenvalues

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Repeated Eigenvalues

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Nonhomogeneous Linear Systems

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Series Solutions for Linear Equations

Certain differential equations lend themselves to approximation through power series. In addition, Euler's equation is a versatile tool to also approximate certain differential equations. Learn to apply these methods around both ordinary and singular points.

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Ordinary Points and Clarification of Singular Points

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Ordinary and singular points

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Series Solutions Near an Ordinary Point

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Series Solutions Near a Regular Singular Point

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Nonlinear Systems

Much like second order differential equations, nonlinear systems are difficult, if not impossible, to solve. Solutions must often be approximated using computers. Explore autonomous systems of equations, the method of linearization to solve them, and the unique cases of conservative systems.

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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

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Autonomous Systems and the Phase Plane

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Autonomous systems

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Stability of Autonomous Systems

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Linearization

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Lyapunov's method

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Laplace Transforms

Given the difficult nature of solving many differential equations, the Laplace Transform is a powerful tool, second only to Fourier. Learn about the Laplace Transform in this theme and apply it to solve ordinary differential equations.

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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

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Review of Partial Fraction Decomposition

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Forward and Inverse Laplace Transforms

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Forward Laplace transforms

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Inverse Laplace transforms

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Solutions of ODEs Using Laplace Transforms

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Step Functions and Impulse

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Step functions

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Impulse/delta functions

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Equations with Discontinuous and Impulsive Forcing

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