Strange constants in desolve_system result for first-order linear system

desolve_system

I'm trying to find the general solution of a first-order linear system with desolve_system. I solved it first with initial conditions, and then wanted to find the general solution:

var('t')
y=function('y')(t)
x=function('x')(t)
print(desolve_system([diff(x,t) == y, diff(y,t) == -10*x + -2*y], [x,y], ics=[0,-1,1]))
print(desolve_system([diff(x,t) == y, diff(y,t) == -10*x + -2*y], [x,y]))

The output is:

[x(t) == -cos(3*t)*e^(-t), y(t) == (cos(3*t) + 3*sin(3*t))*e^(-t)]
[x(t) == 1/3*((x(0) + 1)*sin(3*t) + 3*cos(3*t)*x(0))*e^(-t), y(t) == -1/3*((10*x(0) + 1)*sin(3*t) - 3*cos(3*t))*e^(-t)]

Try it on SageMathCell

In the general solution, there ought to be two unknown constants x(0) and y(0), but instead there's only x(0), and a very strange-looking x(0) + 1.

If I omit the preceding desolve_system with initial values, then I get the expected output

[x(t) == 1/3*((x(0) + y(0))*sin(3*t) + 3*cos(3*t)*x(0))*e^(-t), y(t) == -1/3*((10*x(0) + y(0))*sin(3*t) - 3*cos(3*t)*y(0))*e^(-t)]

Try it on SageMathCell

This is reproducible on SageMathCell and also with SageMath version 10.4, Release Date: 2024-07-19 on MacOS arm64 installed from the Homebrew cask.

Update: Still reproducible with SageMath 10.5. I have now reported it on Github as #39104.

Comments

At Sagecell (running Sage 10.2) and at my local Sage 10.5beta, result is

[x(t) == 1/3*((x(0) + y(0))*sin(3*t) + 3*cos(3*t)*x(0))*e^(-t),
 y(t) == -1/3*((10*x(0) + y(0))*sin(3*t) - 3*cos(3*t)*y(0))*e^(-t)]

Max Alekseyev ( 0 years ago )

I updated the test case. It seems that an earlier command had an effect.

Nate Eldredge ( 0 years ago )

It looks like a bug. Please report at https://github.com/sagemath/sage/issues

Max Alekseyev ( 0 years ago )

This may or may not be a bug :

sage: reset()
sage: t=var("t")
sage: x=function("x")(t)
sage: y=function("y")(t)
sage: des=[diff(x,t) == y, diff(y,t) == -10*x + -2*y]
sage: SolG=desolve_system([diff(x,t) == y, diff(y,t) == -10*x + -2*y], [x,y]) ;
....: SolG
[x(t) == 1/3*((x(0) + 1)*sin(3*t) + 3*cos(3*t)*x(0))*e^(-t),
 y(t) == -1/3*((10*x(0) + 1)*sin(3*t) - 3*cos(3*t))*e^(-t)]

This strange-looking solution does check the original equations :

sage: list(map(lambda c:bool(reduce(lambda a, b:a.substitute_function(b.lhs().operator(), b.rhs().function()), SolG, des[0])), des))
[True, True]

But this solution may be only part of the solution.

To be continued...

Emmanuel Charpentier ( 0 years ago )

Follow-up.

Sympy offers a more complete solution :

sage: import sympy
sage: SolGS=[u._sage_() for u in sympy.dsolve(*map(sympy.sympify, ([diff(x,t) ==
....:  y, diff(y,t) == -10*x + -2*y], [x,y])))] ; SolGS
[x(t) == 1/10*(3*C1 - C2)*cos(3*t)*e^(-t) + 1/10*(C1 + 3*C2)*e^(-t)*sin(3*t),
 y(t) == C2*cos(3*t)*e^(-t) - C1*e^(-t)*sin(3*t)]

which does check the original differential equations:

sage: var("C1, C2")
(C1, C2)
sage: list(map(lambda c:bool(reduce(lambda a, b:a.substitute_function(b.lhs().op
....: erator(), b.rhs().function()), SolGS, des[0])), des))
[True, True]

Emmanuel Charpentier ( 0 years ago )

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