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

• 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 $f(t,y)$ is Lipschitz continuous in $y$ and continuous in $t$, then the initial value problem $y'=f(t,y)$, $y(t_0)=y_0$ has a unique solution on some interval containing $t_0$

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 $f(t,y)$ is said to satisfy the Lipschitz condition with respect to $y$ on a region $R$ if there exists a constant $L$ such that $|f(t,y_1)-f(t,y_2)| \leq L|y_1-y_2|$ for all $(t,y_1)$ and $(t,y_2)$ in $R$
  • The constant $L$ is called the Lipschitz constant
  • Intuitively, this means the function is limited in how quickly it can change with respect to changes in $y$
• 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 $y'=f(t,y)$, $y(t_0)=y_0$, where $t_0$ is the initial point and $y_0$ 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 $y'=y^2+1$, $y(0)=0$. The function $f(t,y)=y^2+1$ is Lipschitz continuous in $y$ and continuous in $t$, so the Picard-Lindelöf theorem guarantees a unique solution exists on some interval containing $t=0$

Interval of Existence

• The interval of existence is the largest interval containing the initial point $t_0$ 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 $y'=y^2$, $y(0)=1$, the solution is $y(t)=\frac{1}{1-t}$, which is defined for all $t<1$. Thus, the interval of existence is $(-\infty,1)$