Differential Equations· Study notes
Analysis
Ordinary differential equations: first- and second-order methods, systems, Laplace transforms, series and Frobenius solutions, qualitative theory, Fourier series and numerical methods.
What you'll be able to do
- Solve linear ODEs with constant coefficients and forcing
- Analyze systems by eigenvalues and phase portraits
- Use Laplace transforms for discontinuous forcing
- Judge stability without solving
Read the notes as a guided lesson
Walk the notes one box at a time — prose, worked examples, quick checks — with a timed practice round after every section. For the full course with interactive lessons, go back to the course.
Every test ends with a score by topic and a list of what to revisit. Test out of a topic with 80% or better to mark it mastered.
Topics
Exact Equations & Substitutions
6 lessonsExactness tests, integrating factors, Bernoulli and homogeneous types
Second-Order Linear Equations
5 lessonsCharacteristic roots: oscillate, decay, or both
The Wronskian & Variation of Parameters
5 lessonsIndependence certificates, and the method that handles any forcing
Nonhomogeneous Equations & Resonance
5 lessonsUndetermined coefficients, and what forcing does
Systems of Differential Equations
5 lessonsEigenvalues run the phase portrait
Nonlinear Systems & Linearization
5 lessonsJacobians, predator–prey, competition, and the pendulum
Laplace Transforms
5 lessonsTurning calculus into algebra
Laplace II: Steps, Impulses & Convolution
5 lessonsPiecewise forcing, hammer blows, and the convolution theorem
Series Solutions
5 lessonsPower series where formulas fail
Frobenius & Special Functions
5 lessonsSingular points, indicial equations, Bessel and Legendre
Models & Qualitative Behavior
5 lessonsGrowth, cooling, mixing, and equilibria
Existence, Uniqueness & Bifurcations
5 lessonsPicard's theorem, phase lines with parameters, harvesting collapse
Boundary Values & Fourier Series
5 lessonsEigenvalue problems, sine series, the heat and wave equations
Numerical Methods: Euler to Runge–Kutta
5 lessonsHeun, RK4, error orders, and stability ceilings