Exam coverage

1. Introduction to Differential Equations
   1. Basic Examples. Solutions of Differential Equations. Interval of Existence (1.1)
   2. Initial Value Problems. Existence-Uniqueness Theorem (1.2)
   3. Differential Equations and Mathematical Models (1.3)
2. First-Order Differential Equations
   1. Solutions Curves without a Solution (2.1)
   2. Separable equations (2.2)
   3. Linear equations (2.3)
   4. Exact Equations. Integrating factors (2.4)
   5. Solutions by substitution: Homogeneous, Bernoulli, linear substitutions (2.5)
   6. A Numerical Method (2.6)
3. Modeling with First-Order Differential Equations
   1. Linear Models (3.1)
   2. Nonlinear Models (3.2)
   3. Modeling with Systems of First-Order ODE (3.3)
4. Higher-Order Differential Equations
   1. Linear equations: Basic Concepts (homogeneous/nonhomogeneous equations, linear superposition principle; existence/uniqueness; Wronskian) (4.1)
   2. Reduction of Order (4.2)
   3. Homogeneous Equations with Constant Coefficients (4.3)
   4. Non-Homogeneous Equations: Method of Undetermined Coefficients (4.4)
   5. Variation of Parameters (4.6)
   6. Cauchy-Euler Equation (4.7)
   7. Nonlinear Equations of Higher Order (4.10)
5. Modeling with Higher-Order Differential Equations
   1. Linear Models: Initial-Value Problems (5.1)
   2. Linear Models: Boundary-Value Problems (5.2)
6. Series Solutions of Differential Equations (Chapter 6)
   1. Review of Power Series (6.1)
   2. Solutions about Ordinary Points (6.2)
   3. Solutions about Regular Singular Points: Method of Frobenius (6.3)
7. The Laplace Transform
   1. The Laplace transform: Integral definition, basic examples (7.1)
   2. Properties of the Laplace transfrom (7.1-3)
   3. Inverse Laplace transform; the partial fractions method (7.2)
   4. Laplace transform and Differential equations: linear equations of first and second order (7.2)
   5. Unit step function, Laplace transform of piecewise functions (7.2)
   6. The Dirac delta function (infinite pulse) (7.5)