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If a second-order differential equation has a characteristic equation with complex conjugate roots of the form r 1 = a + bi and r 2 = a − bi, then the general solution is accordingly y(x) = c 1 e (a + bi )x + c 2 e (a − bi )x. By Euler's formula, which states that e iθ = cos θ + i sin θ, this solution can be rewritten as follows:
The telegrapher's equations then describe the relationship between the voltage V and the current I along the transmission line, each of which is a function of position x and time t: = (,) = (,) The equations themselves consist of a pair of coupled, first-order, partial differential equations. The first equation shows that the induced voltage is ...
The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.
Hill's equation is an important example in the understanding of periodic differential equations. Depending on the exact shape of f ( t ) {\displaystyle f(t)} , solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially. [ 3 ]
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
One millionth of a second. Symbol is μs millisecond: 10 −3 s: One thousandth of a second. Shortest time unit used on stopwatches. jiffy (electronics) ~ 10 −3 s: Used to measure the time between alternating power cycles. Also a casual term for a short period of time. centisecond: 10 −2 s: One hundredth of a second. decisecond: 10 −1 s ...
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
A master equation may be used to model a set of chemical reactions when the number of molecules of one or more species is small (of the order of 100 or 1000 molecules). [4] The chemical master equation can also solved for the very large models, such as the DNA damage signal from fungal pathogen Candida albicans. [5]