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🪝Ordinary Differential Equations

First-order differential equations are the foundation of ODE study. Existence and uniqueness theorems tell us when we can be sure a solution exists and is the only one. This knowledge is crucial for understanding and solving these equations.

These theorems rely on conditions like Lipschitz continuity. They help us determine if a solution exists locally or globally, and over what interval. This information guides our approach to solving and analyzing differential equations.

Existence and Uniqueness Theorems

Fundamental Theorems

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  • Existence theorem proves there is at least one solution to a given differential equation with specified initial conditions
  • Uniqueness theorem proves the solution to a given differential equation with specified initial conditions is unique, meaning there are no other solutions
  • Picard-Lindelöf theorem, also known as the Cauchy-Lipschitz theorem, states that if a function f(t,y)f(t,y) is Lipschitz continuous in yy and continuous in tt, then the initial value problem y=f(t,y)y'=f(t,y), y(t0)=y0y(t_0)=y_0 has a unique solution on some interval containing t0t_0

Local and Global Existence

  • Local existence refers to the existence of a solution to a differential equation in a neighborhood around a specific point or initial condition
    • Proved by the Picard-Lindelöf theorem under certain conditions
    • Guarantees a unique solution exists for at least some small interval around the initial point
  • Global existence refers to the existence of a solution to a differential equation over the entire domain of interest
    • Requires extending the local solution to the maximum possible interval
    • May not always be possible, as solutions can blow up or become undefined at certain points (singularities)
    • Proving global existence often involves showing the solution remains bounded and well-behaved as the independent variable approaches the boundaries of the domain

Conditions for Existence and Uniqueness

Lipschitz Condition

  • A function f(t,y)f(t,y) is said to satisfy the Lipschitz condition with respect to yy on a region RR if there exists a constant LL such that f(t,y1)f(t,y2)Ly1y2|f(t,y_1)-f(t,y_2)| \leq L|y_1-y_2| for all (t,y1)(t,y_1) and (t,y2)(t,y_2) in RR
    • The constant LL is called the Lipschitz constant
    • Intuitively, this means the function is limited in how quickly it can change with respect to changes in yy
  • The Lipschitz condition is a sufficient condition for the existence and uniqueness of solutions to initial value problems
    • It ensures the function is well-behaved enough to guarantee a unique solution
    • Many common functions, such as polynomials and trigonometric functions, satisfy the Lipschitz condition on appropriate regions

Initial Value Problems

  • An initial value problem (IVP) consists of a differential equation along with an initial condition specifying the value of the solution at a particular point
    • For a first-order differential equation, the IVP takes the form y=f(t,y)y'=f(t,y), y(t0)=y0y(t_0)=y_0, where t0t_0 is the initial point and y0y_0 is the initial value
  • The existence and uniqueness theorems apply specifically to initial value problems
    • They guarantee, under certain conditions, that a unique solution to the IVP exists on some interval containing the initial point
  • Example: Consider the IVP y=y2+1y'=y^2+1, y(0)=0y(0)=0. The function f(t,y)=y2+1f(t,y)=y^2+1 is Lipschitz continuous in yy and continuous in tt, so the Picard-Lindelöf theorem guarantees a unique solution exists on some interval containing t=0t=0

Interval of Existence

  • The interval of existence is the largest interval containing the initial point t0t_0 on which a unique solution to an IVP exists
    • It may be a finite interval, a half-infinite interval, or the entire real line, depending on the specific problem
  • The existence and uniqueness theorems guarantee a solution exists on some interval, but do not specify the size of that interval
    • To find the interval of existence, one must often extend the local solution to the maximum possible domain
  • Example: For the IVP y=y2y'=y^2, y(0)=1y(0)=1, the solution is y(t)=11ty(t)=\frac{1}{1-t}, which is defined for all t<1t<1. Thus, the interval of existence is (,1)(-\infty,1)


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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.