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As we saw in Nonhomogenous Linear Equations, differential equations such as this have solutions of the form \[x(t)=c_1x_1(t)+c_2x_2(t)+x_p(t), \nonumber \] where \(c_1x_1(t)+c_2x_2(t)\) is the general solution to the complementary equation and \(x_p(t)\) is a particular solution to the nonhomogeneous equation.
May 11, 2023 · Newton’s second law states that force equals mass times acceleration. So the system of equations governing the setup is \[\begin{align}\begin{aligned} m_1x''_1 &= k(x_2 - x_1) \\ m_2x''_2&= -k(x_2 - x_1) \end{aligned}\end{align} onumber \] In this system we cannot solve for the \(x_1\) or \(x_2\) variable separately.
- Introduction
- Event Detection and Location
- Event Actions
- Events and Discontinuous Differential Equations
- Examples
Differential equations alone are very effective for modeling continuous behavior of systems. However, many real systems involve components that change at discrete times, possibly triggered by states of the continuous solutions. For example, in a heating system, a thermostat will switch on once the temperature reaches a certain level. An event trigg...
Event location works in effectively two stages. In the event detection stage, for each time step taken by the numerical integration method each event is tested to see if it may have changed during the step. Once a possible event is detected, the location stage typically uses a root-finding procedure to find the actual event time .
With WhenEvent[event,action], once an event trigger point is detected and located along a solution trajectory, then the event handler evaluates the action given in the second argument of WhenEvent. First, the solution values are approximated, typically using a dense output formula for the time integration method. The derivatives are found by evalua...
The methods built into NDSolvefor solving systems of differential equations are all effectively based on an implicit assumption that the differential equation satisfies some basic continuity conditions. For ODEs, a sufficient condition is Lipschitz continuity. When a differential equation has discontinuities, these assumptions are violated and solv...
Falling Body
This system models a body falling under the force of gravity encountering air resistance (see [M04]).
Period of the Van der Pol Oscillator
The Van der Pol oscillator is an example of an extremely stiff system of ODEs. The event locator method can call any method for actually doing the integration of the ODE system. The default method, Automatic, automatically switches to a method appropriate for stiff systems when necessary, so that stiffness does not present a problem. By selecting the endpoint of the NDSolve solution, it is possible to write a function that returns the period as a function of . Of course, it is easy to general...
Poincaré Sections
Using Poincarésections is a useful technique for visualizing the solutions of high-dimensional differential systems. For an interactive graphical interface see the package EquationTrekker.
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Nov 16, 2022 · Here is an example of a system of first order, linear differential equations. \[\begin{align*}{{x'}_1} & = {x_1} + 2{x_2}\\ {{x'}_2} & = 3{x_1} + 2{x_2}\end{align*}\] We call this kind of system a coupled system since knowledge of \(x_{2}\) is required in order to find \(x_{1}\) and likewise knowledge of \(x_{1}\) is required to find \(x_{2}\).
x = c1u1(t) + c2u2(t) + + cnun(t) + xp(t). In this expression, each assignment of the constants c1, . . . , cn produces a solution of the nonhomogeneous system, and conversely, each possible solution of the nonhomogeneous system is obtained by a unique special-ization of the constants c1, . . . , cn.
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Aug 27, 2022 · and from T2 at the rate of. (lb/gal in T2) × (3~gal/min) = 1 300Q2 × 3 = 1 100Q2 lb/min. Therefore. (rate in)1 = 5 + 1 100Q2. Solution leaves T1 at the rate of 8 gal/min, since 6 gal/min are drained and 2 gal/min are pumped to T2; hence, (rate out)1 = ( lb/gal in T1) × (8~gal/min) = 1 100Q1 × 8 = 2 25Q1.
Nov 16, 2022 · If \(W e 0\) then the solutions form a fundamental set of solutions and the general solution to the system is, \[\vec x\left( t \right) = {c_1}{\vec x_1}\left( t \right) + {c_2}{\vec x_2}\left( t \right) + \cdots + {c_n}{\vec x_n}\left( t \right)\]
