Hallar la transformada de Laplace

1. 6sen3t-7cos8t

>> syms t;
>> Ft=6*sin(3*t)-7*cos(8*t);
>> fs=laplace(Ft)
fs =18/(s^2 + 9) - (7*s)/(s^2 + 64)

2. 2ch3t+4sh3t

>> syms t;
>> Ft=2*cosh(3*t)+4*sinh(3*t);
>> fs=laplace(Ft)
fs =(2*s)/(s^2 - 9) + 12/(s^2 - 9)

3. (t²+3)²

>> syms t;
>> Ft= (t^2+3)^2;
>> fs=laplace(Ft) 
fs = 9/s + 12/s^3 + 24/s^5

4. t⁴e^2t

>> syms t;
>> Ft=t^4*exp(2*t);
>> fs=laplace(Ft)
fs =24/(s - 2)^5

5. e^-tcos3t

>> syms t;
>> Ft=exp(-t)*cos(3*t);
>> fs=laplace(Ft) 
fs = (s + 1)/((s + 1)^2 + 9) 

6. e^tcos³2t

>> syms t;
>> Ft=exp(t)*(cos(2*t))^3;
>> fs=laplace(Ft)
fs =(s^3 - 3*s^2 + 31*s - 29)/(((s - 1)^2 + 4)*((s - 1)^2 +   36))

7. e^-5t(2sh2t-6cht)

>> syms t;
>> Ft=exp(-5*t)*(2*sinh(2*t)-6*cosh(t));
>> fs=laplace(Ft)  
fs =4/((s + 5)^2 - 4) - (6*(s +  5))/((s + 5)^2 - 1)       

8. ${\displaystyle \frac{\mathrm{sen}t}{t}}$

>> syms t;
>> Ft=sin(t)/t;
>> fs=laplace(Ft) 
fs = atan(1/s)

9. ${\displaystyle \frac{\cos bt-\cos at}{2t}}$

>> syms t a b;
>> Ft=(cos(b*t)-cos(a*t))/(2*t);
>> fs=laplace(Ft)
fs = log(a^2/s^2 + 1)/4 - log(b^2/s^2 + 1)/4

${\displaystyle \frac{1}{4}\ln \left(\frac{s^2+a^2}{s^2+b^2}\right)}$

10. ${\displaystyle \frac{1-\cos 2t}{4t}}$

>> syms t;
>> Ft=(1-cos(2*t))/(4*t);
>> fs=laplace(Ft) 
fs = log(s^2 + 4)/8 - log(s)/4

11. t²cos2t

>> syms t;
>> Ft=t^2*cos(2*t);
>> fs=laplace(Ft) 
fs =(8*s^3)/(s^2 + 4)^3 - (6*s)/(s^2 + 4)^2
>> simplify(fs) 
ans = (2*s*(s^2 - 12))/(s^2 + 4)^3 

12. (t²-2t+5)sent

>> syms t;
>> Ft=(t^2-2*t+5)*sin(t);
>> fs=laplace(Ft)
fs = (5*s^4 - 4*s^3 + 16*s^2 - 4*s + 3)/(s^6 + 3*s^4 + 3*s^2 + 1)
>> simplify(fs)
ans = (5*s^4 - 4*s^3 + 16*s^2 - 4*s + 3)/(s^2 + 1)^3

13. sen³t·ch2t

>> syms t;        
>> Ft=cosh(2*t)*sin(t)^3;
>> fs=laplace(Ft)
fs =3/(((s - 2)^2 + 1)*((s - 2)^2 + 9)) + 3/(((s + 2)^2 + 1)*((s + 2)^2 + 9))
>> pretty(fs)



14. e^-2tsen²t·cos⁴t

Tal y como está no obtiene la solución. Cambiamos

${\displaystyle \begin{array}{l}{\mathrm{sen}}^{2}t{\cos }^{4}t=\frac{1}{16}(1+\cos 2t-\cos 4t-\cos 4t\cos 2t)=\\ \frac{1}{16}\left(1+\frac{1}{2}\cos 2t-\cos 4t-\frac{1}{2}\cos 6t\right)\end{array}}$

>> syms t;
>>  Ft=(1+cos(2*t)/2-cos(4*t)-cos(6*t)/2)/16;
>> fs=laplace(Ft) 
fs = s/(32*(s^2 + 4)) - s/(16*(s^2 + 16)) - s/(32*(s^2 + 36)) +  1/(16*s)
>> simplify(fs)
ans = (2*(s^4 + 28*s^2 + 72))/(s*(s^2 + 4)*(s^2 + 16)*(s^2 + 36))
>> pretty(ans)
    4       2 
2 (s  + 28 s  + 72) 
------------------------------ 
      2        2         2 
 s (s  + 4) (s  + 16) (s  + 36)

Sin descomponer el producto:

>> syms t;
>> Ft=(1+cos(2*t)-cos(4*t)-cos(4*t)*cos(2*t))/16;
>> fs=laplace(Ft)
fs = s/(16*(s^2 + 4)) - (s^3 + 20*s)/(16*(s^4 + 40*s^2 + 144)) - 
s/(16*(s^2 + 16)) + 1/(16*s)
>> simplify(fs)
ans = (2*(s^4 + 28*s^2 + 72))/(s*(s^2 + 4)*(s^2 + 16)*(s^2 + 36))

Luego aplicar la propiedad de traslación.

Calcular las integrales

15. ${\displaystyle {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{e^{-t}(1-\cos 2t)}{2t}}dt}$

>> syms t;
>> Ft=(1-cos(2*t))/(2*t);
fs=laplace(Ft)
fs = log(s^2 + 4)/4 - log(s)/2
>> subs(fs,s,1)
ans = 0.4024

16. ${\displaystyle {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{e^{-t}\mathrm{sen}t}{t}}dt}$

>> syms t;
>> Ft=sin(t)/t;
>> fs=laplace(Ft)
fs = atan(1/s)
>> subs(fs,s,1)
ans = 0.7854
>> pi/4
ans = 0.7854

17. ${\displaystyle {\displaystyle \underset{0}{\overset{\infty }{\int }}{t}^3{e}^{-t}\mathrm{sen}t}·dt}$

>> syms t;
>> Ft= t^3*sin(t);
>> fs=laplace(Ft)
fs = (48*s^3)/(s^2 + 1)^4 - (24*s)/(s^2 + 1)^3 
>> subs(fs,s,1)
ans =0 

Transformada inversa

18. ${\displaystyle L^{-1}\left[\frac{2s-7}{4s^2+25}\right]}$

>> syms s;
>> fs=(2*s-7)/(4*s^2+25);
>> Ft=ilaplace(fs)
Ft =cos((5*t)/2)/2 - (7*sin((5*t)/2))/10

19. ${\displaystyle L^{-1}\left[\frac{5s+3}{9s^2-49}\right]}$

>> syms s;
>> fs=(5*s+3)/(9*s^2-49);
>> Ft=ilaplace(fs)
Ft =13/(63*exp((7*t)/3)) + (22*exp((7*t)/3))/63

20. ${\displaystyle L^{-1}\left[\frac{s}{{\left(s+2\right)}^5}\right]}$

>> syms s;
>> fs=s/(s+2)^5;
>> Ft=ilaplace(fs)
Ft = t^3/(6*exp(2*t)) - t^4/(12*exp(2*t))

21. ${\displaystyle L^{-1}\left[\frac{e^{-4s}}{{\left(s-1\right)}^4}\right]}$

>> syms s;
>> fs=exp(-4*s)/(s-1)^4;
>> Ft=ilaplace(fs)
Ft = (heaviside(t - 4)*exp(t - 4)*(t - 4)^3)/6

Función escalonada unitaria de Heaviside:

${\displaystyle {\rm H}(t-a)=\{\begin{array}{l}1t>a\\ 0t<a\end{array}}$

22. ${\displaystyle L^{-1}\left[\frac{e^{-3s}(s+1)}{s^2+s+1}\right]}$

>> syms s;
>> fs=exp(-3*s)*(s+1)/(s^2+s+1);
>> Ft=ilaplace(fs)
Ft =heaviside(t - 3)*exp(3/2 - t/2)*(cos((3^(1/2)*(t - 3))/2) 
- (3^(1/2)*sin((3^(1/2)*(t - 3))/2))/3) + (2*3^(1/2)*sin((3^(1/2)*(t - 3))/2)
*heaviside(t - 3)*exp(3/2 - t/2))/3

23. ${\displaystyle L^{-1}\left[\frac{3s}{s^2+4}\right]}$

>> syms s;
>> fs=3*s/(s^2+4);
>> Ft=ilaplace(fs)
Ft =3*cos(2*t)

24. ${\displaystyle L^{-1}\left[\ln \left(1+\frac{1}{s^2}\right)\right]}$

>> syms s;
>> fs=log(1+s^-2);
>> Ft=ilaplace(fs)
Ft =-(2*(cos(t) - 1))/t

25. ${\displaystyle L^{-1}\left[\frac{1}{s}\ln \left(1+\frac{1}{s^2}\right)\right]}$

>> syms s;
>> fs=(log(1+s^-2))*s^-1; 
>> Ft=ilaplace(fs) 
Ft = ilaplace(log(1/s^2 + 1)/s, s, t)

26. ${\displaystyle L^{-1}\left[\frac{1}{s^3(s^2+9)}\right]}$

>> syms s;
>> fs=1/(s^3*(s^2+9));
>> Ft=ilaplace(fs)
Ft =cos(3*t)/81 + t^2/18 - 1/81
 
>> fs=s^-3*(s^2+9)^-1;
>> Ft=ilaplace(fs) 
Ft =cos(3*t)/81 + t^2/18 - 1/81

27. ${\displaystyle L^{-1}\left[\frac{1}{{\left(s^2+9\right)}^2}\right]}$

>> syms s;
>> fs=(s^2+9)^-2;
>> Ft=ilaplace(fs)
Ft =sin(3*t)/54 - (t*cos(3*t))/18

28. ${\displaystyle L^{-1}\left[\frac{1}{s^2(s+5)}\right]}$

>> syms s;
>> fs=s^-2*(s+5)^-1;
>> Ft=ilaplace(fs)
Ft =t/5 + 1/(25*exp(5*t)) - 1/25

29. ${\displaystyle L^{-1}\left[\frac{s+5}{s^2-2s-3}\right]}$

>> syms s;
>> fs=(s+5)/(s^2-2*s-3);
>> Ft=ilaplace(fs) 
Ft = 2*exp(3*t) - 1/exp(t)

30. ${\displaystyle L^{-1}\left[\frac{3s+5}{(s+1)(s-2)(s+4)}\right]}$

>> syms s;
>> fs=(3*s+5)/((s+1)*(s-2)*(s+4));
>> pretty(fs)
                        3 s + 5 
                ----------------------- 
               (s + 1) (s - 2) (s + 4)

>> Ft=ilaplace(fs) 
Ft=(11*exp(2*t))/18 - 2/(9*exp(t)) - 7/(18*exp(4*t))

31. ${\displaystyle L^{-1}\left[\frac{3s^2+2s+4}{{\left(s-2\right)}^2(s+4)}\right]}$

>> syms s;
>> fs=(3*s^2+2*s+4)/((s-2)^2*(s+4));
>> Ft=ilaplace(fs)
Ft =(16*exp(2*t))/9 + 11/(9*exp(4*t))+(10*t*exp(2*t))/3

32. ${\displaystyle L^{-1}\left[\frac{s^2+2s-4}{(s-2)(s^2+4)}\right]}$

>> syms s;
>> fs=(s^2+2*s-4)/((s-2)*(s^2+4));
>> Ft=ilaplace(fs)
Ft =cos(2*t)/2 + exp(2*t)/2 + (3*sin(2*t))/2

33. ${\displaystyle L^{-1}\left[\frac{s+1}{{\left(s^2+2s+2\right)}^2}\right]}$

>> syms s;
>> fs=(s+1)/(s^2+2*s+2)^2;
>> Ft=ilaplace(fs) 
Ft =(t*sin(t))/(2*exp(t))

34. ${\displaystyle L^{-1}\left[\frac{s^2+3s-1}{(s^2+2s+5)(s^2+2s+2)}\right]}$

>> syms s;
>> fs=(s^2+3*s-1)/((s^2+2*s+5)*(s^2+2*s+2));
>> Ft=ilaplace(fs)
Ft = (cos(t)-4*sin(t))/(3*exp(t))-(cos(2*t)- (7*sin(2*t))/2)/(3*exp(t))
>> pretty(Ft)



35. ${\displaystyle L^{-1}\left[\frac{s^2}{{\left(s^2+16\right)}^2}\right]}$

>> syms s;
>> fs=s^2/(s^2+16)^2;
>> Ft=ilaplace(fs)
Ft =sin(4*t)/8 + (t*cos(4*t))/2

36. ${\displaystyle L^{-1}\left[\frac{s}{{\left(s^2+4\right)}^3}\right]}$

>> syms s;
>> fs=s/(s^2+4)^3;
>> Ft=ilaplace(fs) 
Ft = (t*sin(2*t))/64 - (t^2*cos(2*t))/32

37. ${\displaystyle L^{-1}\left[\frac{s^3+16s-24}{s^4+20s^2+64}\right]}$

>> syms s;
>> fs=(s^3+16*s-24)/(s^4+20*s^2+64);
>> Ft=ilaplace(fs) 
Ft = cos(2*t) - sin(2*t) + sin(4*t)/2

38. ${\displaystyle L^{-1}\left[\frac{1}{s}\ln \left(\frac{s^2+a^2}{s^2+b^2}\right)\right]}$

>> syms s;
>> fs=log((s^2+a^2)/(s^2+b^2));
>> Ft=ilaplace(fs)
Ft =(2*cos(b*t))/t - (2*cos(a*t))/t

La solución del ejercicio es la integral de 0 a t de Ft

39. ${\displaystyle L^{-1}\left[\mathrm{arctg}\left(\frac{1}{s}\right)\right]}$

>> syms s;
>> fs=atan(1/s);
>> Ft=ilaplace(fs) 
Ft =sin(t)/t

40. ${\displaystyle L^{-1}\left[\ln \left(\frac{s+5}{s+3}\right)\right]}$

>> syms s;
>> fs=log((s+5)/(s+3));
>> Ft=ilaplace(fs) 
Ft =1/(t*exp(3*t)) - 1/(t*exp(5*t))

Ecuaciones diferenciales

1. ${\displaystyle y\text{'}\text{'}+2y\text{'}+5y=e^{-t}\mathrm{sen}t,y(0)=0,y\text{'}(0)=1}$

>> syms y;
>> y=dsolve('D2y+2*Dy+5*y=exp(-t)*sin(t)','y(0)=0','Dy(0)=1')
y =sin(t)/(6*exp(t)) + sin(2*t)/(6*exp(t)) - sin(3*t)/(8*exp(t))
 + sin(5*t)/(24*exp(t)) + (sin(2*t)*(cos(t)/4 - cos(3*t)/12 + 1/6))/exp(t)
>> simplify(y) 
ans = (sin(t)*(2*cos(t) + 1))/(3*exp(t))
>> pretty(ans)
  sin(t) (2 cos(t) + 1) 
  --------------------- 
        3 exp(t)

${\displaystyle y=\frac{1}{3}{e}^{-t}(\mathrm{sen}t+\mathrm{sen}2t)}$

2. ${\displaystyle y\text{'}\text{'}+4y\text{'}+4y=8e^{-2t},y(0)=1,y\text{'}(0)=1}$

>> syms y;
>> dsolve('D2y+4*Dy+4*y=8*exp(-2*t)','y(0)=1','Dy(0)=1')
ans =exp(-2*t) + 3*t*exp(-2*t) + 4*t^2*exp(-2*t)
>> simplify(ans)
ans =exp(-2*t)*(4*t^2 + 3*t + 1)

>> syms s;
>> fs=(8/(s+2)+s+5)/(s^2+4*s+4);
>> ilaplace(fs)
ans =exp(-2*t) + 3*t*exp(-2*t) + 4*t^2*exp(-2*t)

${\displaystyle y=4t^2{e}^{-2t}+3t{e}^{-2t}+e^{-2t}=e^{-2t}(4t^2+3t+1)}$