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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
laplace(R*(1-exp(-t/T)),t,s);
                                    1     1
(%o1)                            R (- - -----)
                                    s   1
                                        - + s
                                        T
(%i2) ilt(s * R * (1/s-1/(1/T+s)),s,t);
                                       - t/T
                                   R %e
(%o2)                              ---------
                                       T
(%i3) laplace(sin(w*t)*sin(w*t+phi),t,s);
                                     2
                         2 cos(phi) w  + sin(phi) s w
(%o3)                    ----------------------------
                                      2    3
                                 4 s w  + s
(%i4) 
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) 

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