I haven't used sympy's Laplace transform code but I can tell you that
dsolve is able to solve the differential equation:
In [14]: y = Function('y')
In [15]: t = Symbol('t')
In [16]: g = 8*t*(Heaviside(t) - Heaviside(t-5)) + 40*Heaviside(t-5)
In [17]: eq = Eq(y(t).diff(t, 2) + 4*y(t), g)
In [18]: str(eq)
Out[18]: 'Eq(4*y(t) + Derivative(y(t), (t, 2)), 8*t*(Heaviside(t) -
Heaviside(t - 5)) + 40*Heaviside(t - 5))'
In [19]: str(dsolve(eq))
Out[19]: 'Eq(y(t), 2*(t*Heaviside(t) + 5*Heaviside(t - 5))*cos(2*t)**2
+ (C1 - 10*cos(10)*Heaviside(t - 5) +
4*Integral(t*sin(2*t)*Heaviside(t - 5), t))*cos(2*t) +
2*(t*Heaviside(t) - t*Heaviside(t - 5) + 5*Heaviside(t -
5))*sin(2*t)**2 + (C2 - cos(2*t)*Heaviside(t - 5) - Heaviside(t) +
cos(10)*Heaviside(t - 5))*sin(2*t))'
There is an unevaluated integral in there and it's quite slow because
of computing the integrals involved so there can be improvements to
made to the integration routines here.
Not having used sympy's laplace transforms before I'm not sure how
they would be used for a case like this:
In [24]: s = Symbol('s')
In [25]: str(laplace_transform(eq.lhs - eq.rhs, t, s))
Out[25]: '(4*LaplaceTransform(y(t), t, s) +
LaplaceTransform(Derivative(y(t), (t, 2)), t, s) - 40*exp(-5*s)/s +
8*(5*s - exp(5*s) + 1)*exp(-5*s)/s**2, 0, True)'
I guess that `LaplaceTransform(Derivative(y(t), (t, 2)), t, s)` is an
unevaluated Laplace transform of the derivative. How exactly does
Matlab represent that?
Oscar
On Wed, 13 Jan 2021 at 18:46, Staffan Lundberg
<[email protected]> wrote:
>
>
> I am working with a project to replace Matlab with Python, in a calculus
> course. Explicitly, following problem is solved in Matlab. It is an
> inhomogenous IVP with constant coefficients together with relevant IC's. In
> Matlab Code: Dy=diff(y(t),t); D2y=diff(Dy,t);
> g=8*t*(heaviside(t)-heaviside(t-5))+40*heaviside(t-5); ode=D2y+4*y-g ==0; My
> goal is to Laplace transform the equation, obtain a solution in frequency
> domain, and finally transform it back to time domain, obtaining the final
> solution. This procedure is successfully done by Matlab.
>
> I address this issue to the developer team. Will it, in future releases of
> SymPy, be possible to solve this problem in Python/SymPy?
>
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