### The Maxima on-line user's manual

Algebra Calculator

#### Ilt

Function: ilt (<expr>, <s>, <t>) Computes the inverse Laplace transform of <expr> with respect to <s> and parameter <t>. <expr> must be a ratio of polynomials whose denominator has only linear and quadratic factors. By using the functions `laplace` and `ilt` together with the `solve` or `linsolve` functions the user can solve a single differential or convolution integral equation or a set of them.

```          (%i1) integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
t
/
[                                    2
(%o1)        I  f(t - x) sinh(a x) dx + b f(t) = t
]
/
0
(%i2) laplace (%, t, s);
a laplace(f(t), t, s)   2
(%o2)  b laplace(f(t), t, s) + --------------------- = --
2    2           3
s  - a           s
(%i3) linsolve ([%], [laplace(f(t), t, s)]);
2      2
2 s  - 2 a
(%o3)     [laplace(f(t), t, s) = --------------------]
5         2     3
b s  + (a - a  b) s
(%i4) ilt (rhs (first (%)), s, t);
Is  a b (a b - 1)  positive, negative, or zero?```

pos; sqrt(a b (a b - 1)) t 2 cosh(---------------------) 2 b a t

`          (%o4) - ----------------------------- + -------`
`                        3  2      2               a b - 1`
`                       a  b  - 2 a  b + a`

2 + ------------------ 3 2 2 a b - 2 a b + a

There are also some inexact matches for `ilt`. Try `?? ilt` to see them.

```(%o1)                                true
(%i2) ```

### Related Examples

##### ilt-invert-matrix

A:matrix([8+5*s,4],[4...

invA:invert(matrix([8...

E:matrix([24/s],[24/s]);

Calculate

##### ilt-plot2d

f(t):=ilt( (s*(0.0012...

f(t);

plot2d(f(t),[t,0,1]);

Calculate

##### ilt

f(p):=2/(p*(p -1) *(p...

ilt(f(p), p, t);

Calculate

##### ilt

ilt((Kp*s+Ki)/s, s, t);

Calculate

##### ilt

ilt(1/(k*s+1)/s,s,t);

Calculate

ilt(s^0.5,s,t);

Calculate

##### ilt

f=ilt(s/((s^2+1)*((s-...

g=ilt((s-2)/(s^2-2*s+...

Calculate

##### ilt

ilt(((s+2)^2)/s^3,s,t);

ilt((1)/(5*s-2),s,t);

ilt((2*s-6)/((s^2)+9)...

Calculate

##### ilt

F(s)=L[f(t)];

f:t;

F:Laplace*(f,t,s);

Calculate

##### ilt-plot2d

f: ilt((1)/s^2, s, t);

plot2d(f, [t, 0, 5]);

Calculate