DiffEq

Existence and Uniqueness Theorem

States that for first order ODEs the initial value problem dy/dx = f(x,y), y(x₀) = y₀ will have a unique solution if f and df/dy are continuous in some rectangle R = {(x,y): a <x<b, c <y<d} containing (x₀,y₀)

An IVP with an nth order differential equation is given by

F(x,y, dy/dx, …, dⁿy/dxⁿ) = 0 with initial condition y(x₀) = y₀, y’(0) = y_n, y^(n-1)(x₀) = y_n-1

An implicit solution yields

more than one explicit solutions

We say the function φ(x) is an explicit soltuion to a differential equation over an interval I if

when substituted for y (dependent variable) it solves the differential equation for all x in I