Computer Algebra Recipes

116 CHAPTER 3. LINEAR ODE MODELS
( t (t¡2)
IF := (1 + t) e 2 )
With the integrating factor known, the standard mathematical procedure is to
multiply ode by IF and integrate. This is easily accomplished with the ¯rst
integral (firint) command.
> sol1:=firint(ode*IF);
μ
t (t¡2) t (t¡2)
sol1 := e
( ) 2 + e
( ) 2 t v(t) ¡ 3 te (1=2 t2 ¡ 2 t) ¡ 4 e (1=2 t2 ¡ 2 t)
+ 5
2 I p ¼e (¡2) p μ
1
2erf
2 I p 2 t ¡ p
2 I ¡ t2 e (1=2 t2 ¡ 2 t) + C1 =0
In the solution output, I stands for p ¡1, erf for the error function, 3 and C1
is the integration constant to be determined. Then, sol1 is solved for v(t), the
result (assigned the name V ) not being shown here in the text.
> V:=solve(sol1,v(t));
Evil Knievel's initial velocity is zero. So the constant C1 is determined by
evaluating the velocity V at t = 0, equating the result to 0, and solving for
C1 . The constant will be automatically substituted into V .
> _C1:=solve(eval(V,t=0)=0,_C1);
C1 := 4 + 5
2 I p ¼e (¡2) p 2erf( p 2 I)
On simplifying V , Evil Knievel's velocity is now determined at arbitrary time
t ¸ 0. This is a nontrivial result to derive by hand.
> Vel:=simplify(V);
Vel := 1
μ
6 te
2
+2t 2 e
t (t¡4)
( 2
t (t¡4)
( 2
) +8e (
t (t¡4)
2 ) p (¡2) ¡ 5 I ¼e p μ
1
2erf
) p
¡ 8 ¡ 5 I ¼e (¡2) p 2erf( p
2 I) e
2 I p 2 t ¡ p
2 I
t (t¡2)
(¡ 2
) ± (1 + t)
Now, since the original ODE was completely real, the velocity must also be
completely real. It can be converted to a real form using the Re (real part of)
command, assuming that t>0. The new function er¯ is the imaginary error
function, 4 which is a real function here.
> Vel:=Re(Vel) assuming t>0;
Vel := 1
Ã
6 te
2
+2t 2 e
t (t¡4)
( 2
t (t¡4)
( 2
R
3 2 u
De¯ned as erf(u) = p¼
) +8e (
t (t¡4)
2 ) p (¡2) +5 ¼e p à p
2 t
2er¯
) p (¡2) ¡ 8+5 ¼e p 2er¯( p
2) e
0 e¡t2
dt:
4 De¯ned as er¯(u) =¡I erf(Iu)= 2
p¼
R u
0 et2
dt.
t (t¡2)
(¡ 2
2 ¡ p 2
!
) ± (1 + t)