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Solutions and Initial Value Problems.
The Approximation Method of Euler.
Introduction: Motion of a Falling Body.
Special Integrating Factors.
Substitutions and Transformations.
Heating and Cooling of Buildings.
Improved Euler's Method.
Higher-Order Numerical Methods: Taylor and Runge-Kutta.
Introduction: The Mass-Spring Oscillator.
Homogeneous Linear Equations; the General Solution.
Auxiliary Equations with Complex Roots.
Nonhomogeneous Equations: the Method of Undetermined Coefficients.
The Superposition Principle and Undetermined Coefficients Revisited.
Variation of Parameters.
Qualitative Considerations for Variable-Coefficient and Nonlinear Equations.
A Closer Look at Free Mechanical Vibrations.
A Closer Look at Forced Mechanical Vibrations.
Interconnected Fluid Tanks.
Elimination Method for Systems with Constant Coefficients.
Solving Systems and Higher-Order Equations Numerically.
Introduction to the Phase Plane.
Coupled Mass-Spring Systems.
Dynamical Systems, Poincaré Maps, and Chaos.
Basic Theory of Linear Differential Equations.
Homogeneous Linear Equations with Constant Coefficients.
Undetermined Coefficients and the Annihilator Method.
Method of Variation of Parameters.
Introduction: A Mixing Problem.
Definition of the Laplace Transform.
Properties of the Laplace Transform.
Inverse Laplace Transform.
Solving Initial Value Problems.
Transforms of Discontinuous and Periodic Functions.
Impulses and the Dirac Delta Function.
Solving Linear Systems with Laplace Transforms.
Introduction: The Taylor Polynomial Approximation.
Power Series and Analytic Functions.
Power Series Solutions to Linear Differential Equations.
Equations with Analytic Coefficients.
Cauchy-Euler (Equidimensional) Equations.
Method of Frobenius.
Finding a Second Linearly Independent Solution.
Review 1: Linear Algebraic Equations.
Review 2: Matrices and Vectors.
Linear Systems in Normal Form.
Homogeneous Linear Systems with Constant Coefficients.
Nonhomogeneous Linear Systems.
The Matrix Exponential Function.
Introduction: A Model for Heat Flow.
Method of Separation of Variables.
Fourier Cosine and Sine Series.
The Heat Equation.
The Wave Equation.
Method of Least Squares.
Runge-Kutta Precedure for n Equations.