Differential Equations

First Order Linear Differential Equations

For a simple equation , the solution can be computed by directly taking the integral .

A separable equation is of the form , where having some antiderivative ​# Differential Equations and gives the rearrangement , solving

Given a continuous function , a general first order autonomous system is , while a first order non-autonomous system is . Additionally, solutions of this equation are called the integral curves of the vector field

In general, a first order linear equation is of the form . After finding the antiderivative , a general solution can be found

Furthermore, so long as and are continuous over the bound , the above solution is unique given a fixed

Bernoulli Equations

An equation of the following form is a Bernoulli equation.

By taking a substitution , this can be solved with some computation.

Logistic Equation

In order to model the growth of a population, we can use a logistic equation , where is a proportionality factor related to population.

Ricatti Equations

An equation of the following from is a Ricatti equation

This equation cannot be explicitly solved, but given an existing solution , we can come up with a general solution of the form .

This is a Bernoulli equation with . Solving as such, we can find a solution for . A useful guess for is .

Reduction of Order

For a second order equation, if it only depends on , then the substitution allows for a simple solution.

In the more case that it depends on instead of , take the same definition of . Then let and . A solution would satisfy .

We let be a function of . From the chain rule, we have

Autonomous

Maximal Interval for is one where monotone on and

For , Barrow’s Formula gives

Taking the limit as goes to endpoints of , if they are defined, then solution is not unique and will reach equilibrium point eventually.

If is Lipschitz, then any solution is unique and exists for all time on the maximum interval.

For a flow transform of some autonomous DE,

Non-Autonomous

Lipschitz continuity also implies uniqueness for non autonomous DEs

Equilibrium

For , some where is an equilibrium point, and a solution is a steady state solution.

Flow Transformation

For a given IVP , a solution where has a corresponding flow transformation . These can be composed, with

By differentiating the flow transform with respect to , we can get the sensitivity function, which tells us how much the solution will change as we vary the initial conditions. For a FOLDE, this would be

This can be also defined for an autonomous system .

Linear Systems

A linear vector field is one that can be represented by a matrix .

A general solution can be represented as a superposition of linear independent solutions. Suppose we have eigenvectors of , then taking A solution with the initial position is then

Furthermore, we can define a flow on this solution

For a complex solution , these are two unique solutions and .

If all the eigenvalues are strictly negative, the solution will converge to , otherwise it may go to infinity.

Duhamel’s Formula

Stability

For equilibrium point of vector field . It is Lyapunov stable if for every , there is a where solutions with , there is a solution for all where .

An equilibrium point is asymptotically stable if there is a where all solutions with have .

The key difference here is that Lyapunov stability allows for solutions to remain a fixed distance away from the equilibrium point, while asymptotically stable means it will reach the equilibrium point at the limit.

When the eigenvalues have strictly negative real parts, then the equilibrium is asymptotically stable