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Substitute Values Symbolically 28 July 2004
subs( x=r^(1/3), 3*x*ln(x^3) );

Evaluate Absolute Value Symbolically 28 July 2004
evalc(abs(I*w*C*R/(I*w*C*R+1)));

Solve Symbolically 28 July 2004
> restart;
> solve( (w*t)/sqrt((w*t)^2+1)*sqrt(2)=1,t);

Plotting Graphs Example 01. 28 July 2004
restart;
> t:=4;
> plot((w*t)/sqrt((w*t)^2+1),w);

Plotting Graphs Example 02. 28 July 2004
> plot(cos(x) + sin(x), x=0..Pi);
> plot(tan(x), x=-Pi..Pi);
> plot([sin(t), cos(t), t=-Pi..Pi]);
> plot(sin(t),t);

LaPlace Transformation Example 01. 24 July 2004

> restart:
> with(inttrans):
> alias(Y(s)=laplace(y(t),t,s)):
> laplace(D(y)(t), t,s);

s Y(s) - y(0)

> laplace((D@@2)(y)(t), t,s);

s (s Y(s) - y(0)) - D(y)(0)

> laplace((D@@3)(y)(t), t,s);

(2)
s (s (s Y(s) - y(0)) - D(y)(0)) - (D )(y)(0)

> ode1:= (D@@2)(y)(t) + 4*y(t) = exp(t);

(2)
ode1 := (D )(y)(t) + 4 y(t) = exp(t)

> laplace(ode1,t,s);

1
s (s Y(s) - y(0)) - D(y)(0) + 4 Y(s) = -----
s - 1

> subs(y(0)=0, D(y)(0) = 1, %);

2 1
s Y(s) - 1 + 4 Y(s) = -----
s - 1

> solve(%, Y(s));

s
-----------------
3 2
s - s + 4 s - 4

> sol1:= invlaplace(%,s,t);

2
sol1 := - 2/5 cos(t) + 4/5 sin(t) cos(t)
 
+ 2/5 exp(1/2 t) cosh(1/2 t)

 

 

LaPlace Transformation Example 02. 24 July 2004

> restart;

with(inttrans): with(plots):
> ode1:=diff(y(t),t,t)+4*y(t)=cos(t);

/ 2 \
|d |
ode1 := |--- y(t)| + 4 y(t) = cos(t)
| 2 |
\dt /

> Lap1:=laplace(ode1,t,s);

Lap1 := s (s laplace(y(t), t, s) - y(0)) - D(y)(0)

s
+ 4 laplace(y(t), t, s) = ------
2
s + 1

> Lp1:=subs(laplace(y(t),t,s)=Y(s),Lap1);

s
Lp1 := s (s Y(s) - y(0)) - D(y)(0) + 4 Y(s) = ------
2
s + 1

> solve(Lp1,Y(s)): Y(s):=%;

3 2
y(0) s + s y(0) + D(y)(0) s + D(y)(0) + s
Y(s) := -------------------------------------------
4 2
s + 5 s + 4

> soln1:=invlaplace(Y(s),s,t);

soln1 :=

1/3 cos(t) + (- 1/3 + y(0)) cos(2 t) + 1/2 D(y)(0) sin(2 t)