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Question: solving system of odes

Question:solving system of odes

mehdi jafari 769 Maple
ode dsolve differential-equations
December 20 2019
0

I have 4 ode equations. i just want to know can i use any option or simplification to have a analytical solution or NOT? Thanks in Advance

 

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restart:
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ode1 := -2*diff(lambda(t),t)*y1(t) - lambda(t)*diff((y1)(t),t)-0*diff(eta(t),t) - diff((y1)(t),t$3) + diff((y1)(t),t)*(y1(t)^2 + y2(t)^2) +4*y1(t)*sqrt(y1(t)^2 + y2(t)^2)*diff(sqrt(y1(t)^2 + y2(t)^2),t)+diff((y1)(t),t)/r^2
+ y1(t)^2*diff(y1(t),t) + y1(t)*y2(t)*diff(y2(t),t) - 2*diff(y1(t),t)/r^2 ;
> 

 

-2*(diff(lambda(t), t))*y1(t)-lambda(t)*(diff(y1(t), t))-(diff(diff(diff(y1(t), t), t), t))+(diff(y1(t), t))*(y1(t)^2+y2(t)^2)+2*y1(t)*(2*y1(t)*(diff(y1(t), t))+2*y2(t)*(diff(y2(t), t)))-(diff(y1(t), t))/r^2+y1(t)^2*(diff(y1(t), t))+y1(t)*y2(t)*(diff(y2(t), t))

 (1)
> 

ode2 := diff((lambda)(t),t$2) + lambda(t)*(y1(t)^2 + y2(t)^2) - 2*y1(t)*diff((y1)(t),t$2) - y1(t)^2*(y1(t)^2 + y2(t)^2) - y1(t)^2/r^2 - diff((y1)(t),t)^2 - 2*diff(sqrt(y1(t)^2 + y2(t)^2),t)^2 - 2*sqrt(y1(t)^2 + y2(t)^2)*diff(sqrt(y1(t)^2 + y2(t)^2),t$2) - diff((y2)(t),t)^2 - 2*y2(t)*diff((y2)(t),t$2) - y2(t)^2*(y1(t)^2 + y2(t)^2)

diff(diff(lambda(t), t), t)+lambda(t)*(y1(t)^2+y2(t)^2)-2*y1(t)*(diff(diff(y1(t), t), t))-y1(t)^2*(y1(t)^2+y2(t)^2)-y1(t)^2/r^2-(diff(y1(t), t))^2-(1/2)*(2*y1(t)*(diff(y1(t), t))+2*y2(t)*(diff(y2(t), t)))^2/(y1(t)^2+y2(t)^2)-2*(y1(t)^2+y2(t)^2)^(1/2)*(-(1/4)*(2*y1(t)*(diff(y1(t), t))+2*y2(t)*(diff(y2(t), t)))^2/(y1(t)^2+y2(t)^2)^(3/2)+(1/2)*(2*(diff(y1(t), t))^2+2*y1(t)*(diff(diff(y1(t), t), t))+2*(diff(y2(t), t))^2+2*y2(t)*(diff(diff(y2(t), t), t)))/(y1(t)^2+y2(t)^2)^(1/2))-(diff(y2(t), t))^2-2*y2(t)*(diff(diff(y2(t), t), t))-y2(t)^2*(y1(t)^2+y2(t)^2)

 (2)
> 

ode3 := 2*diff((lambda)(t),t)*y2(t) + lambda(t)*diff((y2)(t),t) - y1(t)*y2(t)*diff((y1)(t),t) - 4*y2(t)*sqrt(y1(t)^2 + y2(t)^2)*diff((sqrt(y1(t)^2 + y2(t)^2)),t) - y2(t)^2*diff((y2)(t),t) - (y1(t)^2 + y2(t)^2)*diff((y2)(t),t) - diff((y2)(t),t$3) ;

2*(diff(lambda(t), t))*y2(t)+lambda(t)*(diff(y2(t), t))-y1(t)*y2(t)*(diff(y1(t), t))-2*y2(t)*(2*y1(t)*(diff(y1(t), t))+2*y2(t)*(diff(y2(t), t)))-y2(t)^2*(diff(y2(t), t))-(y1(t)^2+y2(t)^2)*(diff(y2(t), t))-(diff(diff(diff(y2(t), t), t), t))

 (3)
> 

ode4 := lambda(t)*y1(t)/r + mu(t)*r - diff((y1)(t),t$2)/r -1/r*y1(t)*(y1(t)^2 + y2(t)^2) - y1(t)/r^3-2/r*diff(y1(t),t$2)

lambda(t)*y1(t)/r+mu(t)*r-3*(diff(diff(y1(t), t), t))/r-y1(t)*(y1(t)^2+y2(t)^2)/r-y1(t)/r^3

 (4)
> 

sys := [ode1, ode2, ode3, ode4]:
> 

dsolve(sys,[y1(t),y2(t),lambda(t),mu(t)],'implicit')

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