Mathematica solves the differential equation
y'' + y' + x * y = 0
in terms of AiryBi[ ] and AiryAi[ ] functions.
In Maxima 5.18.1, contrib_ode returns a solution which
can be written in factored form as
(bessel_y(1/3, f ) * %k2 + bessel_j(1/3, f ) *%k1 ) * g,
where f is (4*x-1)^(3/2)/12
and g is sqrt(4*x-1) * exp( - x/2 ) .
I would like to show that this explicit function of x
is a solution to the original differential equation by
using the first and second derivatives, but when
expanded I am left with eighteen terms which don't
simplify. (besselexpand:true only helps for orders n/2 with
n odd )
To be able to use Maxima to solve differential equations whose
solutions can be expressed in terms of bessel functions,
we need more simplification tools which know about
bessel functions.
Ted Woollett
ps : code for problem:
(%i1) display2d:false$
(%i2) de : 'diff(y,x,2) + 'diff(y,x) + x*y;
(%o2) 'diff(y,x,2)+'diff(y,x,1)+x*y
(%i3) ode2(de,y,x);
(%o3) false
(%i4) load(contrib_ode)$
(%i5) y : rhs(first(contrib_ode(de,y,x)));
(%o5) bessel_y(1/3,(4*x-1)^(3/2)/12)*%k2*sqrt(4*x-1)*%e^-(x/2)
+bessel_j(1/3,(4*x-1)^(3/2)/12)*%k1*sqrt(4*x-1)*%e^-(x/2)
(%i6) dydx : diff(y,x)$
(%i7) d2ydx : diff(y,x,2)$
(%i8) de_check : expand( d2ydx + dydx +x*y )$
(%i9) length( de_check );
(%o9) 18
(%i10) first(de_check) + second(de_check);
(%o10) bessel_y(7/3,(4*x-1)^(3/2)/12)*%k2*x*sqrt(4*x-1)*%e^-(x/2)/4
+bessel_y(1/3,(4*x-1)^(3/2)/12)*%k2*x*sqrt(4*x-1)*%e^-(x/2)/2