Initial Value Problems

Numerical Analysis · Axiom Academy

Starting from a known point and marching forward An Initial Value Problem (IVP) combines a differential equation with a starting condition. Given the equation and where we begin, we want to find where we'll be at any future time. The initial condition y(t₀) = y₀ is crucial—it picks out one specific solution from the infinite family of curves that satisfy the ODE. Visualizing the Solution Family A first-order ODE has infinitely many solutions. The initial condition selects exactly one: Exponential growth with initial population P₀. Solution: P(t) = P₀e^(rt) Object cools toward ambient temperature T_a. Rate proportional to temperature difference. Velocity under gravity with air resistance. Terminal velocity emerges as equilibrium. Capacitor charging through resistor. Voltage approaches source value exponentially. The Picard-Lindelöf theorem tells us when an IVP has a unique solution: Most well-behaved ODEs satisfy these conditions. Trouble arises with singularities or when f "blows up" too fast. Even when a unique solution exists, computing it analytically may be impossible. Numerical methods give us a way forward: Discretize time: Replace continuous t with discrete steps t₀, t₁, t₂, ... Approximate the derivative: Use finite differences to estimate dy/dt March forward: Compute y₁, y₂, ... from the ODE Control error: Choose step size to balance accuracy and speed

This is the written version of the interactive lesson above. See the full Numerical Analysis course.