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Math-dashti

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These are questions asked by Math-dashti

How convert ODE equation to a system?...

Math-dashti 120 Maple 2024
ode homework differential-equations
June 28 2025
1 3

i want to change tmy ODE equation to systems, this procedure is called galilian transformation as i saw , and this is just for second order or we can used for 3rd order and up too ?

there is a lot of example which i know by hand but i want to write by maple

 

How find stability by polar?...

Math-dashti 120 Maple 2024
ode homework polar
June 28 2025
2 5

in this equation i solved without polar which is not shown well without polar specially when a=0 without polar if we solve this  and the other point are complex which not shown like a point in plot, except point (0,0) other not fine, but i want to solve by polar for finding  limit cycle the polar way are so usefull and also for some nonlinear system how i can  change the system to polar and find r* and theta* and plot them ?

polar.mw

How i can fix this issue ?...

Math-dashti 120 Maple 2024
ode homework differential-equations
June 28 2025
1 1

i want to plot the function appear in P but there is a variable t which appear from variable which my system is depending on u,v and must v appear in my function instead of exponential so exponential contain the t variable  there is any way for fixing this?

> 

restart;
> 

with(plots):

 

> 

f := (u,v) -> -u+u^3;
g := (u,v) -> -2*v;

proc (u, v) options operator, arrow; -u+u^3 end proc

  

proc (u, v) options operator, arrow; -2*v end proc

 (1)

 

> 

equilibria := solve({f(u,v)=0, g(u,v)=0}, {u,v});

{u = 0, v = 0}, {u = 1, v = 0}, {u = -1, v = 0}

 (2)

 

> 

de1 := diff(u(t),t) = f(u(t),v(t));
de2 := diff(v(t),t) = g(u(t),v(t));

diff(u(t), t) = -u(t)+u(t)^3

  

diff(v(t), t) = -2*v(t)

 (3)
   
> 

PDEtools:-ConservedCurrents({de1, de2}, [u(t), v(t)]); P1 := -(1/2)*op(1, rhs(op(%)))

[_J[t](t, u(t), v(t)) = f__1((-u(t)^2+1)*exp(-2*t)/u(t)^2, v(t)*exp(2*t))]

  

-(1/2)*(-u(t)^2+1)*exp(-2*t)/u(t)^2

 (4)
> 

P := -(-u^2+1)*exp(-2*t)/(2*u^2)

-(1/2)*(-u^2+1)*exp(-2*t)/u^2

 (5)

 

> 

equilibria;

{u = 0, v = 0}, {u = 1, v = 0}, {u = -1, v = 0}

 (6)

 

> 

p1 := contourplot(P, u=-1.5..2, v=-1.5..1.5, scaling=constrained,
    colorscheme="DivergeRainbow", contours=[seq](x, x=-0.4..0.4,0.1)):

Error, (in plot/iplot2d:-Levels) could not evaluate expression

  

 

> 

p2 := pointplot([[0,0],[1,0],[-1,0]], symbol=solidcircle, symbolsize=15, color=black):

 

> 

p3 := fieldplot([f(u,v), g(u,v)], u=-1.5..2, v=-1.5..1.5,
        arrows=medium, fieldstrength=fixed(0.4), grid=[10,10], labels=["",""]):

 

> 

display(p1,p2,p3, axes=normal);

 

  

Download p1-1.mw

can stability of one equation shown by ...

Math-dashti 120 Maple 2024
ode differential-equations plotting
June 27 2025
1 0

it is apear by sign of taking derivative from equation and substitute in it the sign less zero is stable otherwise is unstable but i want to shown by graph like shown in graph

plot-stablity-1equation.mw

How remove this sign Root0f- even use a...

Math-dashti 120 Maple 2024
solve implicit allvalues
June 27 2025
0 2

what is problem in here

> 

restart;
> 

with(plots):
> 

with(LinearAlgebra):
> 

with(DEtools):

diff(u(x), x) = f(u(x), v(x)), diff(v(x), x) = g(u(x), v(x)) for the two differential equations.

> 

f := (u,v) -> u+cos(v);
g := (u,v) -> u*v-v+sin(v);

proc (u, v) options operator, arrow; u+cos(v) end proc

  

proc (u, v) options operator, arrow; v*u-v+sin(v) end proc

 (1)

The equilibria:

> 

equilibria := solve({f(u,v)=0, g(u,v)=0}, {u,v},explicit);

{u = 1, v = Pi}, {u = -cos(RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z)), v = RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z)}

 (2)
> 

allvalues(RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z))

RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z, 1.306542374), RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z, -1.306542374), RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z, -2.331122370), RootOf(sin(_Z)*_Z^2+sin(_Z)-2*_Z, 2.331122370), 0

 (3)
 

NULL

Download remove.mw

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