# Use ParametricNDSolve with Plotting

I am attempting to solve a differential equation of the form $$\frac{dx}{dt}=ax+by \\ \frac{dy}{dt}=cx$$ with parameters $0<a<1$ and $a,b,c \in \mathbb{R}$. I am able to use NDParametric Solve, but would like to generate a stream vector plot of the solutions, with $a=\lbrace 0,0.2,0.4,0.6,0.8,1.0 \rbrace$ b and c as the x-axis and y-axis. I can solve for varying values of $a$, but how do I turn this into a stream vector plot?

• Could you please post your code for the NDParametric Solve? Feb 28 '15 at 18:24
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• @user21807 for a, b, c constants the system is linear and has a simple analytic solution. Why not take that? Feb 28 '15 at 21:10

As has been observed by Dr. Wolfgang Hintze if a,b and c are constant this coupled system has easily derived analytic solutions. In general, this site strongly encourages you to try to present your attempts and then focused clear guidance can be given. I post this as in this case this does not really require more than 'out of the box' functions. Perhaps it can be illustrative and motivate your own play:

Manipulate[
Show[StreamPlot[{a x + b y, c y}, {x, -3, 3}, {y, -3, 3}],
ParametricPlot[
Evaluate[{x[t], y[t]} /.
First@DSolve[{{x'[t], y'[t]} == {{a, b}, {0, c}}.{x[t], y[t]},
x[0] == ix, y[0] == iy}, {x[t], y[t]}, t]], {t, 0, 10},
PlotStyle -> Red] /. Line[w__] :> Arrow[w],
Epilog -> {Red, PointSize[0.02], Point[{ix, iy}]}], {a, 0.01,
1}, {b, 0, 1}, {c, 0, 1}, {ix, -3, 3}, {iy, -3, 3}]


I have not aimed for efficiency, terseness or clarity just coded on the run to motivate starting...

• How do you go about creating that movie (GIF I suppose)? Mar 6 '15 at 2:33
• @Amzoti apologies for delay in replying. I use licecap.en.softonic.com to screen capture. You can can create animated gifs from Mathematica. There are a number of helpful searchable posts on this site. Mar 7 '15 at 3:01
• No problem - excellent +1, thanks. Mar 7 '15 at 3:06