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laplace

Run Example
(%i1)assume(V>
0, R>
0, L>
0, C>
0);
(%o1)                    [V > 0, R > 0, L > 0, C > 0]
(%i2) assume(C*R^2 - 4*L >
 0);
                                     2
(%o2)                            [C R  > 4 L]
(%i3) ec  : R*x(t) + L*diff(x(t),t) + integrate(x(t), t)/C = V;
                                             /
                                             [
                                             I x(t) dt
                                             ]
                              d              /
(%o3)               x(t) R + (-- (x(t))) L + --------- = V
                              dt                 C
(%i4) ecd : diff(ec, t);
                                     2
                    d               d               x(t)
(%o4)              (-- (x(t))) R + (--- (x(t))) L + ---- = 0
                    dt                2              C
                                    dt
(%i5) e2:laplace(ecd,t,s);
                                                     !
                                            d        !
(%o5) (s laplace(x(t), t, s) - x(0)) R + (- -- (x(t))!
                                            dt       !
                                                     !t = 0
                    2                                   laplace(x(t), t, s)
                 + s  laplace(x(t), t, s) - x(0) s) L + ------------------- = 0
                                                                 C
(%i6) atvalue( x(t), t=0, 0);
(%o6)                                  0
(%i7) atvalue( diff(x(t), t), t=0, V/L);
                                       V
(%o7)                                  -
                                       L
(%i8) e4:solve(e2,laplace(x(t),t,s));
                                                   !
                                          d        !
                              x(0) C R + (-- (x(t))!      + x(0) s) C L
                                          dt       !
                                                   !t = 0
(%o8)  [laplace(x(t), t, s) = -----------------------------------------]
                                                  2
                                         s C R + s  C L + 1
(%i9) ilt(e2,s,t);
(%o9) ilt((s laplace(x(t), t, s) - x(0)) R, s, t)
                   !
          d        !         2
 + ilt((- -- (x(t))!      + s  laplace(x(t), t, s) - x(0) s) L, s, t)
          dt       !
                   !t = 0
       laplace(x(t), t, s)
 + ilt(-------------------, s, t) = 0
                C
(%i10) 
Run Example
defint(x,x,a-b,a);
                              2    2            2
                             a    b  - 2 a b + a
(%o1)                        -- - ---------------
                             2           2
(%i2) eq1:sin(w*x);
(%o2)                              sin(w x)
(%i3) eq2:defint(eq1,x,a-b,a);
                           cos((a - b) w)   cos(a w)
(%o3)                      -------------- - --------
                                 w             w
(%i4) laplace(eq2,a,s);
                     w sin(b w) + s cos(b w)        s
(%o4)                ----------------------- - -----------
                               2    2              2    2
                           w (w  + s )         w (w  + s )
(%i5) 
Run Example
Gc:0;
(%o1)                                  0
(%i2)  Rg:0;
(%o2)                                  0
(%i3)  Ro1:0;
(%o3)                                  0
(%i4)  Ro2:0;
(%o4)                                  0
(%i5) v(t):=0;
(%o5)                              v(t) := 0
(%i6) i(t):=IL;
(%o6)                             i(t) := IL
(%i7) atvalue(vc(t), t=0, VG);
(%o7)                                 VG
(%i8) atvalue(ig(t), t=0, IL);
(%o8)                                 IL
(%i9) atvalue(i1(t), t=0, IL);
(%o9)                                 IL
(%i10) atvalue(i2(t), t=0, 0);
(%o10)                                 0
(%i11) atvalue(vut1(t), t=0, 0);
(%o11)                                 0
(%i12) atvalue(iut1(t), t=0, 0);
(%o12)                                 0
(%i13) vFW1(t):=-VG;
(%o13)                          vFW1(t) := - VG
(%i14) iFW1(t):=0;
(%o14)                           iFW1(t) := 0
(%i15) vFW2(t):=0;
(%o15)                           vFW2(t) := 0
(%i16) atvalue(vut2(t), t=0, 0);
(%o16)                                 0
(%i17) eqi1:laplace(ig(t) = Gc*vc(t) + C*'diff(vc(t),t,1) + i(t),t,s);
                                                                   IL
(%o17)    laplace(ig(t), t, s) = C (s laplace(vc(t), t, s) - VG) + --
                                                                   s
(%i18) eqi2:laplace(i(t) + iFW1(t) = i1(t),t,s);
                           IL
(%o18)                     -- = laplace(i1(t), t, s)
                           s
(%i19) eqi3:laplace(Gc*vc(t) + C*'diff(vc(t),t,1) + iut1(t) = iFW1(t),t,s);
(%o19)   C (s laplace(vc(t), t, s) - VG) + laplace(iut1(t), t, s) = 0
(%i20) eqv1:laplace(VG = Rg*ig(t) + v(t) + Ro1*i1(t) + Lo1*'diff(i1(t),t,1) + Lu*'diff(i2(t),t,1),t,s);
       VG
(%o20) -- = Lo1 (s laplace(i1(t), t, s) - IL) + Lu s laplace(i2(t), t, s)
       s
(%i21) eqv2:laplace(vut1(t) = vFW1(t) + Ro1*i1(t) + Lo1*'diff(i1(t),t,1) + Lu*'diff(i2(t),t,1),t,s);
                                  VG
(%o21) laplace(vut1(t), t, s) = - -- + Lo1 (s laplace(i1(t), t, s) - IL)
                                  s
                                                    + Lu s laplace(i2(t), t, s)
(%i22) eqv3:laplace(v(t) = vc(t) + vFW1(t),t,s);
                                                    VG
(%o22)                   0 = laplace(vc(t), t, s) - --
                                                    s
(%i23) eqv4:laplace(vut2(t) = vFW2(t) + Ro2*i2(t) + Lo2*'diff(i2(t),t,1) + Lu*'diff(i1(t),t,1),t,s);
(%o23) laplace(vut2(t), t, s) = Lu (s laplace(i1(t), t, s) - IL)
                                                   + Lo2 s laplace(i2(t), t, s)
(%i24) sol:solve([	eqi1,	eqi2,	eqi3,	eqv1,	eqv2,	eqv3,	eqv4],[	'laplace(vc(t),t,s),	'laplace(ig(t),t,s),	'laplace(i1(t),t,s),	'laplace(iFW1(t),t,s),	'laplace(i2(t),t,s),	'laplace(vut2(t),t,s),	'laplace(vFW1(t),t,s)]);
(%o24)                                []
(%i25) 

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