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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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SOLVING LINEAR EQUATION CHEATS
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Hours Topic
4

### Introduction to Differential Equations

• Definitions and terminology
• Initial-value problems
• Differential equations as mathematical models
8

### First-Order Differential Equations

• Solution curves without a solution; direction fields, autonomous first-order differential equations
• Separation of variables
• Linear equations
• Exact equations
• Solutions by substitutions
• Numerical methods; Euler's method, numerical solvers
4

### Modeling with First-Order Differential Equations

• Linear models; exponential growth and decay, Newton's law of cooling, mixture problems
• Non-linear models; logistic growth, chemical reactions
• Systems of differential equations; radioactive series, mixtures, predator-prey models
14

### Higher-Order Differential Equations

• Linear differential equations; initial-value and boundary-value problems, homogenous equations, non-homogeneous equations
• Reduction of order
• Homogenous linear equations with constant coefficients
• Undetermined coefficients; supposition approach, annihilator approach
• Variation of parameters
• Cauchy-Euler equation
• Solving systems on linear equations using elimination
• Non-linear differential equations
5

### Modeling with Higher-Order Differential Equations

• Linear models with initial value problems; spring/mass systems with free undamped motion, free damped motion, and driven motion
• Linear models with boundary value problems
• Nonlinear models
7

### Series Solutions of Linear Equations

• Review of power series
• Special functions; Bessel's Equation, Legendre's Equation
10

### The Laplace Transform

• Definition of the Laplace transform
• Inverse transforms and transforms of derivatives
• Operational properties of the transform
• The Dirac Delta function
6

### Systems of Linear First-Order Differential Equations

• Preliminary theory; supposition principle, independence, general solutions
• Homogenous linear systems; distinct real eigenvalues, repeated eigenvalues, complex eigenvalues
• Nonhomogeneous linear systems; undetermined coefficients, variation of parameters
• Matrix exponentials
4

### Numerical Solutions of Ordinary Differential Equations

• Euler's method
• Improved Euler's method
• Runga-Kutta methods (RK4)