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solution = DSolve[{p'[t] a p[t] b p[t]2, p[0] p0}, p[t],t]
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Solve ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
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a at p0 p[t] a b p0 + b at p0
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population[t_] = solution[[1, 1, 2]]; Plot[population[t]/.{p0 1, a 2, b .05}, {t,0,5}, PlotRange All]
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Limit[population[t] /. {p0 1, a 2, b .05}, t ] 40.
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Ordinary Differential Equations
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11.14 Solve the boundary value problem
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d2y + 4 2 y = 0 , y(0) = y(1) = 0. dx 2
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DSolve[{y''[x] + 4 o2 y[x] 0, y[0] 0, y[1] 0}, y[x], x]
{{y[x] C[2] Sin[2
x]}}
11.2 Numerical Solutions
Although certain types of differential equations can be solved analytically in terms of elementary functions, the vast majority of equations that arise in applications cannot. Even if unique solutions can be shown to exist, it may only be possible to obtain numerical approximations. The command NDSolve is designed specifically for this purpose.
NDSolve[equations, y, {x, xmin, xmax}] gives a numerical approximation to the solution, y, of the differential equation with initial conditions, equations, whose independent variable, x, satisfies xmin x xmax.
Because NDSolve yields a numerical solution to a differential equation, or system of differential equations, an appropriate set of initial conditions that guarantees uniqueness must be specified.
dy = x 2 + y with initial condition y(0) = 1. dx Although this equation has a unique solution, it cannot be found in terms of elementary functions using DSolve.
EXAMPLE 14 In this example we consider the differential equation
DSolve y'[x] x 2 + y[x], y[0] 1 , y[x], x DSolve y'[x] x2 + y[x] ,y[0] 1 ,y[x], x
We can only obtain a numerical approximation to the solution of this equation. Because numerical techniques construct approximations at only a finite number of points, Mathematica interpolates, i.e., constructs a smooth function passing through these points and returns the solution as an InterpolatingFunction object.
EXAMPLE 15
temp = NDSolve {y'[x] x 2 + y[x], y[0] 1}, y,{x,0,1}
InterpolatingFunction[{{0.,1.}}, <> ]}}
The actual interpolating function can now be extracted from this expression:
solution = temp[[1, 1,2]] InterpolatingFunction[{{0.,1.}}, <> ]
Only the domain of an InterpolatingFunction object is printed explicitly. The remaining elements are represented as < >. To see the data used in its construction, enter the command FullForm[solution]. Using the interpolated solution, solution, we can compute the solution at one or more points, and we can even plot it. One must be careful, however, to stay within the domain of the interpolating function or a warning will be generated.
solution[0.5] 1.60643 solution[1.5]
InterpolatingFunction dmval : Input value {1.5} lies outside the range of data in the interpolating function. Extrapolation will be used.
An extrapolated value is not as reliable as an interpolated value, in terms of accuracy.
Ordinary Differential Equations
list = Table[{x, solution[x]}, {x, 0, 1, .1}]; TableForm[list, TableSpacing {1, 5}]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. 1. 1.10284 1.21273 1.33181 1.46228 1.60643 1.76656 1.94504 2.14429 2.36672 2.61479
Plot[solution[x], {x, 0, 1}, AxesOrigin {0,0}]
2.5 2.0 1.5 1.0 0.5 0.2 0.4 0.6 0.8 1.0
Although the default settings for NDSolve work nicely for most differential equations, Mathematica provides some options that can be used to set parameters to handle abnormal situations.
WorkingPrecision is an option that specifies how many digits of precision should be maintained internally in computation. The default (on most computers) is WorkingPrecision 16. AccuracyGoal is an option that specifies how many significant digits of accuracy are to be obtained. The default is AccuracyGoal Automatic, which is half the value of WorkingPrecision. AccuracyGoal effectively specifies the absolute error allowed in a numerical procedure. PrecisionGoal is an option that specifies how many effective digits of precision should be sought in the final result. The default is PrecisionGoal Automatic, which is half the value of WorkingPrecision. PrecisionGoal effectively specifies the relative error allowed in a numerical procedure. MaxSteps is the maximum number of steps to take in obtaining the solution. The default is MaxSteps Automatic, which, for ordinary differential equations, is 10,000. MaxStepSize specifies the maximum size of each step in the iteration. StartingStepSize specifies the initial step size. The default is StartingStepSize Automatic. (Mathematica automatically determines the best step size for the given equation.)
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