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TianyuCheng

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These are questions asked by TianyuCheng

general solution of ODE...

TianyuCheng 40 Maple
ode homework dsolve
March 09 2020
1 1

solve it by Maple , get the following form solution, what's the Int(1,0)

 


 

> 

restart;

 

> 

ODE :=diff(r(t),t)=r(t)*(1-r(t)^2)+mu*r(t)*cos(t)

diff(r(t), t) = r(t)*(1-r(t)^2)+mu*r(t)*cos(t)

 (1)
> 

dsolve(ODE)

r(t) = ((_C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t)))*exp(2*t+2*mu*sin(t)))^(1/2)/(_C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t))), r(t) = -((_C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t)))*exp(2*t+2*mu*sin(t)))^(1/2)/(_C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t)))

 (2)
> 

g := unapply(sqrt((C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t)))*exp(2*t+2*mu*sin(t)))/(C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t))), t)

proc (t) options operator, arrow; ((C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t)))*exp(2*t+2*mu*sin(t)))^(1/2)/(C1+2*(Int((exp(t))^2*(exp(mu*sin(t)))^2, t))) end proc

 (3)
> 

g(0)

1/(C1+2*(Int(1, 0)))^(1/2)

 (4)

``


 

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The phase portrait of ode system...

TianyuCheng 40
ode differential-equations phaseportrait
March 09 2020
1 2

How can I  get the  phase portrait  and several trajectory of the following system(it is in polar coordinates )

r'=r(1 − r^2) + 2rcos(θ),

θ' = 1

 

dimensionless ODE model...

TianyuCheng 40 Maple
ode differential-equations stability
March 03 2020
2 1

For the following system, the parameters D_1,D_2,D_3,D_4,D_5,D_6,S are all positive, How can I get all the  equilibrium  and their stability.

diff(x1,t)=fS;diff(x2,t)=fV;diff(x3,t)=fC;diff(x4,t)=fR;

fS := -D2*x1*x2-D1*x1-x1*x3+S; fV := D2*x1*x2-D4*x2+x1*x3; fC := -D6*x3*x4-D5*x3+x2; fR := D6*x3*x4-x4

Numerical solution of Non-autonomous ODE...

TianyuCheng 40 Maple
ode dsolve numeric
February 29 2020
1 1

How can I use maple to get the numerical solution of this non-autonomous system

sys_ode := diff(x(t), t) = ax(t)-bx(t)y(t)+2sin(t), diff(y(t), t) = -cy(t)+dx(t)y(t)

where a,b,c,d is parameters, they can take any value.

problem with ode ...

TianyuCheng 40 Maple
ode differential-equations phaseportrait
February 04 2020
1 0

why is  the figure  so weird ? How can I get the smooth trajectories of the ode system?


 

with(DEtools)

> 

with(DEtools)
> 

with(plots)
> 

with(plots, implicitplot)

[implicitplot]

 (1)
> 

with(MatrixTools)

Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received MatrixTools

  
> 

with(LinearAlgebra)
> 

with(VectorCalculus)

R := PolynomialRing([x, y])

PolynomialRing([x, y])

 (2)
> 

``

p1 := phaseportrait([diff(x(t), t) = x(t)*(1-x(t)^2-y(t)^2)+y(t)*(y(t)^2+(x(t)^2-1)^2), diff(y(t), t) = y(t)*(1-x(t)^2-y(t)^2)-x(t)*(y(t)^2+(x(t)^2-1)^2)], [x(t), y(t)], t = 0 .. 50, [[x(0) = -3, y(0) = .5], [x(0) = 4, y(0) = 2]], x = -4 .. 4, y = -4 .. 4, dirgrid = [13, 13], stepsize = 0.5e-1, axes = BOXED)

  
> 

p2 := pointplot([[1, 0], [-1, 0], [0, 0]], symbolsize = 20, symbol = solidcircle, color = black)

 
> 

``

 

> 

display(p1, p2)

 

solve({x*(-x^2-y^2+1)+y*(y^2+(x^2-1)^2), y*(-x^2-y^2+1)-x*(y^2+(x^2-1)^2)}, {x, y})

{x = 0, y = 0}, {x = RootOf(_Z^8-6*_Z^6+11*_Z^4-6*_Z^2+2), y = RootOf(_Z^8-6*_Z^6+11*_Z^4-6*_Z^2+2)^5-3*RootOf(_Z^8-6*_Z^6+11*_Z^4-6*_Z^2+2)^3+RootOf(_Z^8-6*_Z^6+11*_Z^4-6*_Z^2+2)}, {x = 1, y = 0}, {x = -1, y = 0}, {x = RootOf(_Z^2-2), y = RootOf(_Z^2+1)}

 (3)

subs({x = .5, y = -.5}, x*(-x^2-y^2+1)+y*(y^2+(x^2-1)^2))

-.15625

 (4)

sys := [x*(-x^2-y^2+1)+y*(y^2+(x^2-1)^2), y*(-x^2-y^2+1)-x*(y^2+(x^2-1)^2)]

[x*(-x^2-y^2+1)+y*(y^2+(x^2-1)^2), y*(-x^2-y^2+1)-x*(y^2+(x^2-1)^2)]

 (5)

A := Jacobian(sys, [x, y])

Matrix(%id = 18446746647299284134)

 (6)

latex(Matrix(2, 2, {(1, 1) = -2, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}))

 \left[ \begin {array}{cc} -2&0\\ \noalign{\medskip}0&0\end {array}
 \right]

 

A1 := subs([x = 1, y = 0], A)

Matrix(%id = 18446746647299319918)

 (7)

A2 := subs([x = -1, y = 0], A)

Matrix(%id = 18446746647176545934)

 (8)

A3 := subs([x = 0, y = 0], A)

Matrix(%id = 18446746647213830142)

 (9)

r1, e1 := Eigenvectors(A1)

Vector[column](%id = 18446746647213811102), Matrix(%id = 18446746647213811582)

 (10)

r2, e2 := Eigenvectors(A3)

Vector[column](%id = 18446746647299291126), Matrix(%id = 18446746647299291246)

 (11)

``


 

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