# 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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## 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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## 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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## 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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## 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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## 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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## 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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