Existence, Uniqueness and Qualitative Behaviour: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 3, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
If the right-hand side of a first-order equation is continuous on a rectangle around the initial point, at least one solution exists on some interval about that point. Continuity alone buys existence but says nothing about uniqueness.
Picard Lindelof theorem
If the right-hand side is continuous and Lipschitz in the dependent variable on a rectangle about the initial point, then a unique solution exists on some interval containing that point. It is the standard existence and uniqueness result.
Lipschitz condition
A bound stating that the change in the right-hand side between two y values is at most a fixed constant times the distance between those y values. It controls how fast nearby solutions can separate.
Sufficient test for Lipschitz
If the partial derivative of the right-hand side with respect to y is continuous on a closed rectangle, it is bounded there, and the mean value theorem converts that bound into a Lipschitz constant.
Failure of uniqueness
When the right-hand side is continuous but not Lipschitz, several solutions may share one initial condition. The classic example is a fractional power of y at the origin, where a constant solution and a rising solution both fit.
Picard iteration
The successive approximation scheme that rewrites the initial value problem as an integral equation and repeatedly substitutes the previous approximation. Under the Lipschitz condition the iterates converge to the unique solution.
Contraction mapping argument
The reasoning behind Picard iteration, in which the integral operator shrinks distances between candidate functions on a short interval, so a fixed point exists and is unique by the Banach fixed point theorem.
Local versus global existence
The standard theorem guarantees a solution only near the initial point. Solutions of nonlinear equations can blow up in finite time, so a locally guaranteed solution need not extend to all values of the independent variable.
Finite time blow-up
The phenomenon in which a solution grows without bound as the independent variable approaches a finite value, ending the interval of existence there. Superlinear nonlinear growth on the right-hand side is the usual cause.
Continuation of solutions
The principle that a solution extends until it either leaves every compact subset of the region where the hypotheses hold or the independent variable reaches the boundary. It converts local existence into a maximal interval.
Global existence for linear equations
If the coefficient and forcing functions of a first-order linear equation are continuous on an interval, the solution of any initial value problem there exists across that entire interval. Linear problems never blow up inside it.
Continuous dependence on data
The property that small changes in the initial value or in the coefficients produce small changes in the solution over a bounded interval. Together with existence and uniqueness it makes the problem well posed.
Well-posed problem
A problem that has a solution, has only one, and depends continuously on its data. All three are required before a numerical answer to a modelled problem can be regarded as meaningful.
Gronwall inequality
A tool bounding a function that is dominated by a constant plus an integral of itself by an exponential expression. It is the standard route to uniqueness proofs and to continuous dependence estimates.
Equilibrium solution
A constant solution of an autonomous equation, obtained by setting the right-hand side to zero. Equilibria organise the whole phase line because no solution can cross one under uniqueness.
Phase line
A one-dimensional diagram marking the equilibria of an autonomous equation and using arrows to record where the right-hand side is positive or negative. It gives the qualitative behaviour of every solution at a glance.
Asymptotically stable equilibrium
An equilibrium that nearby solutions approach as the independent variable increases, shown by arrows pointing toward it on both sides of the phase line. It is often called a sink.
Unstable equilibrium
An equilibrium that nearby solutions move away from on both sides, with phase line arrows pointing outward. Also called a source, it is only ever observed exactly, never approached.
Semistable equilibrium
An equilibrium attracting solutions from one side and repelling them on the other, arising when the right-hand side touches zero without changing sign. A small parameter change usually splits or destroys it.
Linearised stability test
Evaluating the derivative of the right-hand side at an equilibrium of an autonomous equation. A negative value means asymptotic stability, a positive value means instability, and zero leaves the test inconclusive.
Solutions cannot cross
A consequence of uniqueness: two distinct solution curves of the same equation can never meet in the region where the hypotheses hold, since a shared point would give one initial condition two solutions.
Comparison principle
The result that if one right-hand side dominates another everywhere, then solutions of the larger equation starting no lower stay no lower. It bounds hard solutions between easy ones without solving anything.
Bifurcation of equilibria
A qualitative change in the number or stability of equilibria as a parameter passes a critical value. Small parameter shifts can create, destroy or exchange the stability of equilibria in an autonomous model.
Euler method
The simplest numerical scheme, stepping forward by the current slope times the step size. Its global error shrinks in proportion to the step size, so halving the step roughly halves the error.
Improved step accuracy
Schemes that sample the slope more than once per step, such as midpoint or classical fourth-order Runge Kutta methods, achieve far smaller error for the same step size at the cost of extra evaluations.