Differential Equations
Picard–Lindelöf Theorem
A Lipschitz right-hand side guarantees a unique local solution to any ODE initial value problem.
Linear ODE General Solution
The matrix exponential e^{At} gives the unique solution to y' = Ay with initial condition y(0) = y_0.
Integrating Factor Method
Multiplying a first-order linear ODE by e^{∫p dx} converts it into an exact derivative and yields a closed-form solution.
Wronskian
The Wronskian W(f,g) = fg' - f'g detects linear independence of two solutions to a second-order ODE.
Variation of Parameters
A particular solution to a non-homogeneous ODE is built from the homogeneous solutions via Wronskian-weighted integrals.
Heat Equation Maximum Principle
Solutions to u_t = u_xx on a bounded domain attain their maximum on the parabolic boundary, not in the interior.
D'Alembert's Wave Equation Solution
Every solution to u_tt = c²u_xx decomposes into a right-traveling wave and a left-traveling wave.
Laplace's Equation Mean Value Property
A function is harmonic iff it equals the average of its values over any sphere centered at that point.