Computer Algebra Recipes

3.3. SPECIAL FUNCTION MODELS 139
the expression for the tension being automatically substituted.
The equation of motion is a linear partial di®erential equation. Solving linear
PDEs is the subject matter of Chapters 5 and 6. The PDE may be converted
into an ODE by assuming a solution of the form Ã(y;t) =X(y)cos(!t), with
! taken to be a positive angular frequency.
> psi(y,t):=X(y)*cos(omega*t);
Ã(y; t) :=X (y)cos(!t)
With the assumed solution automatically substituted, the resulting output of
eq is divided by cos(!t) and expanded to produce the ODE eq2 .
> eq2:=expand(eq/cos(omega*t));
μ μ μ
2
2
d
d d
eq2 := ¡²g X (y) + ²g X (y) L ¡ ²g X (y) y = ¡² X (y) !
dy dy2 dy2 2
The second derivative terms are collected in eq2 ,
> eq3:=collect(eq2,diff(X(y),y,y));
μ μ 2 d d
eq3 := (²gL¡ ²gy) X (y) ¡ ²g
dy2 dy
and a general analytic solution to eq3 obtained.
X (y) = ¡² X (y) ! 2
> sol:=dsolve(eq3,X(y));
μ r
μ r
L ¡ y
L ¡ y
sol := X (y) = C1 BesselJ 0; 2 ! + C2 BesselY 0; 2 !
g
g
The general solution is a linear combination of zeroth-order Bessel functions of
the ¯rst and second kinds with arbitrary constants C1 and C2 . The Bessel
Y0 function diverges to ¡1 at y = L so must be removed on physical grounds.
> X:=remove(has,rhs(sol),BesselY);
μ r
L ¡ y
X := C1 BesselJ 0; 2 !
g
To remove the arbitrary constant C1 ,theop command is used to select the
second operand in X.
> X:=op(2,X);
μ r
L ¡ y
X := BesselJ 0; 2
Taking the bungee cord length to be L = 30 meters, its density ² = 1
2 kilogram/meter,
and g =9:8 meter/second 2 ,theformofX is as follows:
> L:=30: g:=9.8: epsilon:=1/2: X:=X;
X := BesselJ(0; 2 p 3:061224489 ¡ 0:1020408163 y!)
The transverse displacement X of the cord at y = 0 is zero, which is entered as
a boundary condition, bc.
g
!
> bc:=eval(X,y=0)=0;
bc := BesselJ(0; 3:499271060 !) =0