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load('optvar)$
depends(y,x)$
/*
The functional may include derivatives of higher that first
order. For example, including the small-deflection energy
of bending, shear, and an elastic foundation, the elastic
energy of a nonuniform beam with loading W(X) is the
integral of the following function: */
a(x)*'d(y,x,2)**2 + b(x)*'d(y,x)**2 + c(x)*y**2 + w(x)*y$
/* to get the Euler-Lagrange equation: */
eq:el(%, y, x);
/* If the shear & foundation energy are negligible, as
is often the case, we may solve this differential equation,
for specific A(X) and W(X), by two successive applications
of ODE2. For example: */
first(eq),d,infeval,a(x)=x, b(x)=0, c(x)=0, w(x)=1$
bb:%,'d(y,x,3)='d(dy2,x),'d(y,x,4)='d(dy2,x,2);
dependencies(dy2(x))$
ev(bb,d,eval);
ode2(%, dy2, x);
ode2(subst([%k1=%k3,%k2=%k4,dy2='d(y,x,2)],%), y, x);
/* Now let's impose the boundary conditions
Y='D(Y,X)=0 at X=1, Y=0, 'D(Y,X)=1 at X=2: */
ic2(%, x=1, y=0, 'd(y,x)=0);
ic2(subst([%k3=%k1,%k4=%k2],%), x=2, y=0, 'd(y,x)=1);
/* as another specific beam example: */
eq,eval,a(x)=1,b(x)=2,c(x)=1,w(x)=1$
first(%),d;
/* We cannot treat this as 2 successive second-order
equations, but the function DESOLVE in the SHARE file
DESOLN, already loaded, is applicable to some linear
arbitrary-order constant coefficient equations. First we
must convert the equation so that the dependency of Y is
explicitly indicated: */
convert(%, y, x);
/* DESOLVE requires all boundary conditions to be at
X=0, and it works best if they are specified before calling
the function. However, we may overcome this restriction, as
illustrated by the following example, where the beam has
zero slope and deflection at both ends: */
(atvalue(y(x),x=0,0), atvalue('d(y(x),x),x=0,0),
atvalue('d(y(x),x,2),x=0,%k1), atvalue('d(y(x),x,3),x=0,%k2))$
desolve(%th(2), y(x));
ic2(subst(y(x)=y,%), x=1, y=0, 'd(y,x)=0);
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