First-Order DE 2005
Created : 25-Aug-96
Last Modified : 16-Sep-04
Author : Tom Coleman
Copyright Status : In Public Domain
Solver : DES 2005
Schema : 2005.0
DE is a contraction of differential equation.
The dependent variable here is Y and the independent
variable is X. The equation is
dY/dX = - K * Y
with Y0 being the initial value of Y.
K is known as the rate constant. When the independent
variable is time, the time constant is the reciprocal
of the rate constant.
Tau = 1 / K
When X = Tau, Y = 0.37 * Y0.
The half-time is the time required for Y to go
one-half way from its initial value to its final
value.
t 1/2 = 0.69 * Tau
The exact solution of this equation is
Exact Y = Y0 * exp (-K * X)
The error in the calculated solution is
Error = Exact Y - Calculated Y
2005.0 First-OrderDE Model ------------------------------------
K 0.1 Tau t1/2 ExactY Error ------------------------------------
Y Y 1.0 dY/dX ----------------------------------
Parms Tau INVERT K t1/2 0.69 * Tau Dervs dY/dX - K * Y ExactY EXP ( - K * System.X ) Error ExactY - Y $--------------------------------------Model.ParmsModel.Dervs$ ---------------------------------------
10 2 10 1 10 0.2 ---------------------------------------
First-Order DE 0.4 1 14 1 The equation is dY / dX = - K * Y X is the independent variable, Y is the dependent variable, and Y0 is the initial value of Y. 15.6 1 K is known as the rate constant in the equation dY / dX = - K * Y When the independent variable is time, the time constant (Tau) is the reciprocal of the rate constant. Tau = 1 / K When X = Tau, Y = 0.37 * Y0. The half-time is the time required for Y to go one-half way from its initial value to its final value. t 1/2 = 0.69 * Tau 17.2 1 The equation dY / dX = - K * Y with initial Y0 is one of only a few differential equations that have known exact solutions. The exact solution is Y = Y0 * exp ( -K * X ) 18.8 1 The calculated solution is only approximate. The error in this calculation is the difference between the exact solution and the calculated solution. 2 1 Model.Y 1 3 1 Model.dY/dX 2 4 1 Model.ExactY 1 5 1 Model.Error 3
Rate constant
7 1 Rate Constant 5.5 26 RateConst 0.01 19 0.01 8 0.1 1.6 1 Model.K RateConst 3.0 1 Model.Tau 1 4.0 1 Model.t1/2 1
Right-hand Side ---------------------------------
0.5 37 System.X 1 Box RED 2 1 2.5 28 7 32 6 System.X 0 20 Model.Y Model.ExactY NULL RED Box 0 1 10.0 28 5 32 6 System.X 0 20 Model.dY/dX BLUE -0.1 0 15.5 28 5 32 6 System.X 0 20 Model.Error DOT MAGENTA -0.01 +0.01
End