← All courses y ′ y' y ′ Analysis
Ordinary differential equations: first- and second-order methods, systems, Laplace transforms, series and Frobenius solutions, qualitative theory, Fourier series and numerical methods.
Before this: Calculus , Linear Algebra 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 Start the course
Go through each topic step by step — prose, worked examples, quick checks, and a timed practice round after every section.
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 1. First-Order Equations
5 lessons Separable equations and integrating factors
2. Exact Equations & Substitutions
6 lessons Exactness tests, integrating factors, Bernoulli and homogeneous types
3. Second-Order Linear Equations
5 lessons Characteristic roots: oscillate, decay, or both
4. The Wronskian & Variation of Parameters
5 lessons Independence certificates, and the method that handles any forcing
5. Nonhomogeneous Equations & Resonance
5 lessons Undetermined coefficients, and what forcing does
6. Systems of Differential Equations
5 lessons Eigenvalues run the phase portrait
7. Nonlinear Systems & Linearization
5 lessons Jacobians, predator–prey, competition, and the pendulum
8. Laplace Transforms
5 lessons Turning calculus into algebra
9. Laplace II: Steps, Impulses & Convolution
5 lessons Piecewise forcing, hammer blows, and the convolution theorem
10. Series Solutions
5 lessons Power series where formulas fail
11. Frobenius & Special Functions
5 lessons Singular points, indicial equations, Bessel and Legendre
12. Models & Qualitative Behavior
5 lessons Growth, cooling, mixing, and equilibria
13. Existence, Uniqueness & Bifurcations
5 lessons Picard's theorem, phase lines with parameters, harvesting collapse
14. Boundary Values & Fourier Series
5 lessons Eigenvalue problems, sine series, the heat and wave equations
15. Numerical Methods: Euler to Runge–Kutta
5 lessons Heun, RK4, error orders, and stability ceilings