differential-equations

I am trying to find an example of the title. My first thought was to set something like $$ x'(t) = \begin{cases} 1, x \geq 0 \\ 0, \text{else} \end{cases}$$ But this differential equation doesn't have ...

The previous post looks at the nonlinear pendulum equation and what difference it makes to the solutions if you linearize the equation. If the initial displacement is small enough, you can simply replace sin θ with θ. If the initial displacement is larger, you can improve the accuracy quite a bit by solving the linearized […] The post Closed-form solution to the nonlinear pendulum equation first …

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) Rokhlin’s theorem states that the …

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) The Hitchin-Thorpe inequality stat…

In grad school I specialized in differential equations, but never worked with delay-differential equations, equations specifying that a solution depends not only on its derivatives but also on the state of the function at a previous time. The first time I worked with a delay-differential equation would come a couple decades later when I did […] The post Differential equation with a small delay fi…

Here’s a differential equation from [1] that’s interesting to play with. Even though it’s a nonlinear system, it has a closed-form solution, namely where (a, b, c) is the solution at t = 0 and Δ = 1 + a² + b² + c². The solutions lie on the torus (doughnut). If m and n are coprime integers then the solutions […] The post Differential equation on a doughnut first appeared on John D. Cook .

Students seeing differential equations for the first time expect every equation to have a nice closed-form solution, because up to that point in their education nearly every problem they’ve seen has been contrived to have a nice closed-form solution. Once you resign yourself to the fact that a differential equation will rarely have a closed […] The post Rational solution to Korteweg–De Vries equa…

Differential equations in the complex plane are different from differential equations on the real line. Suppose you have an nth order linear differential equation as follows. The real case If the a‘s are continuous, real-valued functions on an open interval of the real line, then there are n linearly independent solutions over that interval. The […] The post Complex differential equations first a…

Most applied differential equations are second order. This probably has something to do with the fact that Newton’s laws are second order differential equations. Higher order equations are less common in application, and when they do pop up they usually have even order, such as the 4th order beam equation. What about 3rd order equations? […] The post Third order ordinary differential equations fi…

When solving a fixed-constant linear ordinary differential equation where the part of the homogeneous solution is same form as part of a possible particular solution, why do we get the next independent solution in the form of x^n* possible form of part of particular solution?  Show this through an example. See the pdf file ______________________ … Continue reading "Why multiply possible form of p…

One of the classical applied problems in ordinary differential equations is that of finding the time of death of a homicide victim. The estimation of time of death is generally based on the temperature of the body at two times – 1) when the victim is found and 2) then a few hours later. Assuming … Continue reading "Time of death – a classic ODE problem"