Differential Equations Study Guide

First-Order ODEs: Separable and Linear Equations

Master separable equations and integrating factor method; foundational techniques appearing in 20-25% of exam problems.

First-Order ODEs: Exact, Bernoulli, and Substitution Methods

Learn exactness tests, integrating factors, and substitution techniques for nonstandard first-order equations on exams.

Existence, Uniqueness, and Qualitative Analysis

Understand existence-uniqueness theorem, direction fields, isoclines, and Euler's method for solution behavior analysis.

Applications of First-Order ODEs

Model population dynamics, Newton's cooling, mixing problems, and RC/RL circuits; common applied exam problems.

Higher-Order Linear Homogeneous ODEs

Solve constant coefficient equations using characteristic equations; master real, repeated, and complex root cases.

Nonhomogeneous ODEs and Cauchy-Euler Equations

Apply undetermined coefficients and variation of parameters; solve Cauchy-Euler equations appearing frequently on exams.

Applications: Mechanical Vibrations and Resonance

Analyze spring-mass systems, damping, forced oscillations, and resonance; essential physics-based exam applications.

Laplace Transforms: Fundamentals and IVP Solutions

Compute transforms, inverse transforms, and solve IVPs; high-value computational problems on final exams.

Laplace Transforms: Discontinuous and Impulse Functions

Handle Heaviside step functions, Dirac delta, and convolution; tests advanced transform mastery on exams.

Systems of First-Order Linear ODEs

Use eigenvalue methods and matrix exponentials; sketch phase portraits for nodes, saddles, spirals, centers.

Series Solutions and Frobenius Method

Find power series solutions near ordinary and regular singular points; tests analytical technique proficiency.

Boundary Value Problems and Fourier Series

Solve eigenvalue BVPs, compute Fourier series, introduce separation of variables for heat equation problems.