Differential Equations Demystified
by STEVEN G. KRANTZ
| Pages | PDF | 4.9 MB |
Content,
What Is a Differential Equation?
- Introductory Remarks
- The Nature of Solutions
- Separable Equations
- First-Order Linear Equations
- Exact Equations
- Orthogonal Trajectories and Families of Curves
- Homogeneous Equations
- Integrating Factors
- Reduction of Order
- The Hanging Chain and Pursuit Curves
- Electrical Circuits
- Exercises
Second-Order Equations
- Second-Order Linear Equations with Constant Coefficients
- The Method of Undetermined Coefficients
- The Method of Variation of Parameters
- The Use of a Known Solution to Find Another
- Vibrations and Oscillations
- Newton’s Law of Gravitation and Kepler’s Laws
- Higher-Order Linear Equations,
- Coupled Harmonic Oscillators
- Exercises
Power Series Solutions and Special Functions
- Introduction and Review of Power Series
- Series Solutions of First-Order Differential Equations
- Second-Order Linear Equations Ordinary Points
- Exercises
Fourier Series: Basic Concepts
- Fourier Coefficients
- Some Remarks About Convergence
- Even and Odd Functions: Cosine and Sine Series
- Fourier Series on Arbitrary Intervals
- Orthogonal Functions
- Exercises
Partial Differential Equations and Boundary Value Problems
- Introduction and Historical Remarks
- Eigenvalues, Eigenfunctions, and the Vibrating String
- The Heat Equation: Fourier’s Point of View
- The Dirichlet Problem for a Disc
- Sturm–Liouville Problems
- Exercises
Laplace Transforms
- Introduction
- Applications to Differential Equations
- Derivatives and Integrals of Laplace Transforms
- Convolutions
- The Unit Step and Impulse Functions
- Exercises
Numerical Methods
- Introductory Remarks
- The Method of Euler
- The Error Term
- An Improved Euler Method
- The Runge–Kutta Method
- Exercises
Systems of First-Order Equations
- Introductory Remarks
- Linear Systems
- Homogeneous Linear Systems with Constant Coefficients
- Nonlinear Systems: Volterra’s Predator–Prey Equations
- Exercises
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