Computer Algebra Recipes

192 CHAPTER 4. NONLINEAR ODE MODELS
and the integrands F and G are entered.
> F:=y(x); G:=sqrt(1+diff(y(x),x)^2);
s
μ
d
F := y(x) G := 1+
dx y(x)
2
Then FF = F + ¸G is formed.
> FF:=F+lambda*G;
s
μ
d
FF := y(x)+¸ 1+
dx y(x)
2
The EulerLagrange commandisappliedtoFF .
> eq:=EulerLagrange(FF,x,y(x));
8
μ
d
>< ¸
dx
eq := 1+
>:
y(x)
2 μ
2 d
y(x)
dx2 Ã μ
d
1+
dx y(x)
μ
2 d
¸ y(x)
dx2 ! ¡ s
2
(3=2) μ
d
1+
dx y(x)
;
2
s
μ
d
y(x)+¸ 1+
dx y(x)
μ
d
2
dx
¡
y(x)
2
¸
s
μ
d
1+
dx y(x)
9
>=
= K1
2
>;
The ¯rst integral is extracted from eq by selecting the term that has K1 in it,
and removing the brackets that would otherwise appear in ode.
> ode:=select(has,eq,K[1])[];
s
μ
d
ode := y(x)+¸ 1+
dx y(x)
μ
d
2
dx
¡
y(x)
2
¸
s
μ
d
1+
dx y(x)
= K1
2
A general analytic solution to the ¯rst-order nonlinear ode is sought, yielding
two answers that di®er by the sign in front of the square root.
> sol:=dsolve(ode,y(x));
p
sol := y(x) =K1 + ¸2 ¡ C1 2 ¡ x2 +2x C1 ;
p
y(x) =K1 ¡ ¸2 ¡ C1 2 ¡ x2 +2x C1
The rhs of, say, the negative square root result (second one) is chosen.
> Y:=rhs(sol[2]);
p
Y := K1 ¡ ¸2 ¡ C1 2 ¡ x2 +2x C1