Computer Algebra Recipes

4.3. VARIATIONAL CALCULUS MODELS 193
The boundary conditions that Y =0atx =0andatx = a are imposed in bc1
and bc2 , which are then solved for K1 and C1 insol2 .
> bc1:=eval(Y,x=0)=0; bc2:=eval(Y,x=a)=0;
p
bc1 := K1 ¡ ¸2 ¡ C1 2 =0
p
bc2 := K1 ¡ ¸2 ¡ C1 2 ¡ a2 +2a C1 =0
> sol2:=solve(fbc1,bc2g,fK[1],_C1g);
sol2 := f C1 = a
2 ;K1
p
4 ¸2 ¡ a2 =
g
2
Assigning sol2 ,weobtainYin the following form.
> assign(sol2): Y:=Y;
p r
4 ¸2 ¡ a2 Y :=
¡ ¸
2
2 ¡ 1
4 a2 ¡ x2 + xa
To determine ¸, the constraint condition R a
G(y(x) =Y ) dx = L is imposed.
0
> eq2:=simplify(int(eval(G,y(x)=Y),x=0..a))=L;
r
¸
eq2 := ¡
2
¡4 ¸2 + a2 p p
¡4 ¸2 + a2 (¡ln(¡a + ¡4 ¸2 + a2 )
+ln(a + p ¡4 ¸2 + a2 )) = L
The values a =1andL =1:5 are entered, and eq2 numerically solved for ¸,
the search range ¸ =0to0:6being speci¯ed.
> a:=1: L:=1.5: lambda:=fsolve(eq2,lambda=0..0.6);
¸ := 0:5014101096
The ¯nal form of Y is then displayed, and jY j plotted in Figure 4.15.
> Y:=Y; plot(abs(Y),x=0..a,scaling=constrained);
Y := 0:03757789244 ¡ p 0:0014120980 ¡ x2 + x
0.4
0.3
0.2
0.1
0
0.2 0.4 0.6 0.8 1
x
Figure 4.15: Shape of the oxhide strip that maximizes the enclosed area.