Map of content for differential equations — equations that describe how quantities change. The path: foundations → first-order methods → second-order methods → Laplace transforms → systems → stability theory.
Foundations
What a differential equation is and what counts as a solution.
- Differential equation — the general concept.
- Ordinary differential equation — single independent variable.
- Linear ODE — the well-behaved class.
- Initial value problem — ODE plus initial conditions.
- Implicit solution to ODE — when can’t be isolated.
- Superposition principle — linear combinations of homogeneous solutions.
- Mathematical modeling with ODEs — translating physical systems to equations.
Existence and uniqueness
When a solution is guaranteed.
- Existence and uniqueness theorem — for first-order ODEs.
- Existence and uniqueness for systems — vector version.
- Picard iteration — constructive proof and approximation method.
- Slope field — visualization of as direction arrows.
First-order solution methods
The toolbox for .
- Separable equation — when factors as .
- Integrating factor — for first-order linear with variable coefficients.
- Exact equation — when a potential function exists.
- Clairaut’s theorem — the exactness test.
Modeling examples (first-order)
Real-world applications.
- Malthusian model — exponential population growth/decay.
- Logistic model — bounded growth with carrying capacity.
- Equilibrium of an ODE — where the system stops changing.
Second-order linear ODEs
Theory and tools.
- Wronskian — linear independence test.
- Representation theorem — general solution from independent solutions.
- Characteristic equation — for constant-coefficient homogeneous.
- Method of reduction of order — finding a second solution from a first.
Nonhomogeneous methods
Particular solutions.
- Particular solution and complementary solution — the decomposition.
- Method of undetermined coefficients — guess and solve for nice forcings.
- Method of variation of parameters — universal method via integrals.
Vibrations (applications)
Second-order ODEs in physics.
- Mechanical and electrical vibrations — damped mass-spring and RLC circuits.
- Resonance — when forcing matches natural frequency.
Laplace transform
Algebraic technique for solving ODEs.
- Laplace transform — definition and core properties.
- Properties of Laplace transform — derivatives, shifts, linearity.
- Method of Laplace transform — the solution procedure for IVPs.
- Inverse Laplace transform — going back to the time domain.
- Heaviside step function — the unit step.
- Rectangular window function — built from Heavisides.
- Transform of discontinuous functions — handling jumps.
- Convolution integral — products in s-domain ↔ convolution in time.
- Transfer function — system input/output in s-domain.
- Impulse response — system’s response to a delta input.
- Dirac delta function — the unit impulse.
Systems of ODEs
Multiple coupled equations.
- System of first-order linear ODEs — overview and conversion from higher-order.
- Linear independence of vector functions — the matrix-form Wronskian.
- Distinct real eigenvalues case — straightforward case.
- Complex conjugate eigenvalues case — oscillating solutions.
- Repeated eigenvalues case — generalized eigenvectors.
- Algebraic vs geometric multiplicity — when extra structure is needed.
Phase plane and stability
Qualitative behavior of 2D autonomous systems.
- Autonomous system — time-independent right-hand side.
- Linear autonomous system — the planar case.
- Phase plane — visualizing trajectories.
- Phase plane behaviour — six standard equilibrium types.
- Stability of autonomous systems — formal stability definitions.
- Critical point of autonomous system — equilibria and isolated points.
Nonlinear stability methods
When linearization isn’t enough.
- Locally linear system — linearizing around equilibria.
- Lyapunov’s method — energy-function approach to stability.
- Lyapunov function — what makes a good .
Connects to Computer architecture (RC delays in CMOS gates, transient response of circuits) and to Engineering economics (compound interest as the simplest exponential ODE). Linear-algebra prerequisites (eigenvalues, eigenvectors, determinants) underpin much of the systems theory.