Preface
This section demonstrates an application of the power series method in solving first order differential equations.
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Glossary
Examples for first order ODEs solved by Power Series
We believe that the best way to learn a material and techniques is go by examples. Therefore, this section contains a detail exposition of several examples.
Example: Consider the initial value problem
Example: Consider the initial value problem
ProductLog
is another name for Lambert function.
Then we plot its solution using standard Mathematica command
NDSolve
and display the corresponding stream plot along with
the singular line (in red) y = 4/x.
Plot[y[x] /. sol, {x, 0, 1.4}, PlotStyle -> Thick]
stream = ListPlot[ Reap[StreamPlot[{1, y^2/(4 - x*y)}, {x, 0, 2.5}, {y, 0, 10}, EvaluationMonitor :> Sow[{x, y}]]][[-1, 1]]]
hyper = Plot[4/x, {x, 0.3, 2.6}, PlotStyle -> {Thick, Red}, PlotRange -> {{0, 2.5}, {0, 10}}]
Show[hyper, stream]
Stream plot | Solution curve |
The given differential equation has a moving singularity: y = 4/x, indicated on the graph by red curve. Now we seek its solution in power series form:
Example: Consider the initial value problem
- Duan, J.-S. and Rach, R., The degenerate form of the Adomian polynomials in the power series method for nonlinear ordinary differential equations, Journal of Mathematics and System Science, Volume 5, Pages 411--428, doi: 10.17265/2159-5291/2015.10.003
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