- #1
amcavoy
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I know how to solve them in Mathematica, but is there a way I can plot slope fields / integral curves?
yes there isapmcavoy said:I know how to solve them in Mathematica, but is there a way I can plot slope fields / integral curves?
<<Graphics`PlotField`
<<Graphics`Arrow`
sol1=NDSolve[{y'[x]==y[x]-x,y[0]==0.5},y,{x,0,3}];
fsol[x_]:=Evaluate[y[x]/.Flatten[sol1]];
xpt=2.6
xed=2.7
ypt=fsol[2.6]
yed=fsol[2.7]
a1=Graphics[Arrow[{xpt,ypt},{xed,yed}]];
pv=PlotVectorField[{1,y-x},{x,-3,3},{y,-3,3},PlotRange->{{-4,4},{-4,4}},
PlotPoints->25,Axes->True]
pt1=Plot[fsol[x],{x,0,2.7},PlotStyle->{{Thickness[0.01]}}]
Show[{pv,pt1,a1}]
apmcavoy said:I'm not on my home computer now, but I will post it tomorrow most likely. I would have thought Wolfram would have a quicker way to do this, but I guess this is it. Thanks a lot saltydog for the information! I will try that out as soon as possible.
<<Graphics`PlotField`
sol1=NDSolve[{y'[x]==y[x]-x,y[0]==0.5},y,{x,0,3}];
pt1=Plot[Evaluate[y[x]/.sol1]
pv=PlotVectorField[{1,y-x},{x,-3,3},{y,-3,3}]
Show[{pt1,pv}]
apmcavoy said:The longer way worked great. However, I couldn't get the cut-down version to work properly. Any ideas why? Thanks saltydog.
<<Graphics`PlotField`
sol1=NDSolve[{y'[x]==y[x]-x,y[0]==0.5},y,{x,0,3}];
pt1=Plot[Evaluate[y[x]/.sol1,{x,0,3}]
pv=PlotVectorField[{1,y-x},{x,-3,3},{y,-3,3}]
Show[{pt1,pv}]
Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are important because they allow us to model and understand various physical and natural phenomena, such as the motion of objects, population growth, and chemical reactions.
Mathematica is a powerful computational software that can solve differential equations numerically and symbolically. It has built-in functions and algorithms specifically designed for solving differential equations, making it an efficient tool for researchers and scientists.
Yes, Mathematica has the capability to handle systems of differential equations, including ordinary and partial differential equations. It can also solve initial value problems and boundary value problems for systems of any size.
While Mathematica is a powerful tool for solving differential equations, it does have limitations. It may not be suitable for solving highly complex or nonlinear equations, and the accuracy of the solutions may vary depending on the chosen numerical methods and parameters.
Yes, Mathematica has built-in visualization tools that can plot the solutions to differential equations in both 2D and 3D. This allows for a better understanding of the behavior of the solutions and can aid in further analysis and interpretation.