D'Alembert reduction of a linear homogeneous ordinary differential equation
This functionality does not run in MATLAB.
ode::dAlembert(Ly, y(x), v)
ode::dAlembert(Ly, y(x), v) returns the reduced differential equation of Ly using the method of reduction of d'Alembert and the function v. If v is a solution of Ly and u is a solution of the reduced differential equation then v ∈ t u is another solution of Ly.
Consider the following differential equation:
Ly := 2/x^3*y(x)-2/x^2*diff(y(x),x)+1/x*diff(y(x),x$2)+ diff(y(x),x$3)
We easily check that x is a particular solution of Ly:
Then we reduce the equation Ly using this special solution:
R := ode::dAlembert(Ly, y(x), x)
The solutions of the equation R are not too hard to find:
ode::evalOde(R, y(x)=1), ode::evalOde(R, y(x)=1/x^3)
So a basis of solutions of Ly is therefore which can be checked directly:
A homogeneous linear differential equation.
The dependent function of Ly.