Ecuaciones diferenciales lineales con coeficientes constantes

Trabajando con el lenguaje simbólico, Matlab tiene la función dsolve para la resolución de estas ecuaciones.

1. 3y''-2y'-8y=03y''-2y'-8y=0

>> syms y;
>> y=dsolve('3*D2y-2*Dy-8*y=0') 
y = C4*exp(2*t) + C3*exp(-(4*t)/3)

La función simplify simplifica una expresión algebraica (en simbólico)

2. y'''-3y''+3y'-y=0, y(0)=1, y'(0)=2, y''(0)=3

>> y=dsolve('D3y-3*D2y+3*Dy-y=0','D2y(0)=3','Dy(0)=2','y(0)=1') 
y =exp(t) + t*exp(t)

3. y'''-5y''=(7t-4)e^2t

>> y=dsolve('D3y-5*D2y=(7*t-4)*exp(2*t)')
y =C10*exp(5*t) - exp(5*t)*(exp(-3*t)*((7*t)/12 - 13/18) -exp(-5*t)*
(C8/25 + C9 + (C8*t)/5))
>> simplify (y) 
ans = C8/25 + C9 + (13*exp(2*t))/18 - (7*t*exp(2*t))/12 + (C8*t)/5 + 
C10*exp(5*t)
>> pretty(ans)
C8        exp(2 t) 13   t exp(2 t) 7   C8 t
-- + C9 + ----------- - ------------ + ---- + C10 exp(5 t)
25             18            12          5

Con la función pretty de Matlab podemos ver mejor la solución.

${\displaystyle \frac{C8}{25}+C9=A_1;\frac{C8}{5}=A_2;C10=B_1}$

Obtenemos

${\displaystyle y=A_1+A_{2}t+B_1{e}^{5t}+\frac{1}{6}\left(\frac{13}{3}-\frac{7}{2}t\right)e^{2t}}$

4. y''-y'=5sent

La integral general

${\displaystyle y=y_H+y_p=A+B{e}^t+\frac{5}{2}\left(\cos t-\text{sen}t\right)}$

>> y=dsolve('D2y-Dy=5*sin(t)') 
y = C11 + C12*exp(t) + (5*2^(1/2)*cos(t + pi/4))/2
>> expand(y)
ans =C11 + (5*cos(t))/2 - (5*sin(t))/2 + C12*exp(t)

5. y''+4y=7cos(2t)

La integral general

${\displaystyle y=y_H+y_p=A\cos 2t+B\text{sin}2t+\frac{7}{4}t\text{sen}2t}$

>> y=dsolve('D2y+4*y=7*cos(2*t)') 
y =(21*cos(2*t))/32 + (7*cos(6*t))/32 + sin(2*t)*((7*t)/4 + 
(7*sin(4*t))/16) + C3*cos(2*t) - C4*sin(2*t)
>> simplify(y) 
ans = (7*cos(2*t))/8 + (7*t*sin(2*t))/4 + C3*cos(2*t) - C4*sin(2*t)

6. 4y''+8y'=2tsent

${\displaystyle y=A+B{e}^{-2t}+\left(-\frac{1}{25}-\frac{1}{5}t\right)\cos t+\left(\frac{7}{25}-\frac{1}{10}t\right)\text{sen}t}$

>> dsolve('2*D2y+4*Dy=t*sin(t)')
ans =
C5 - cos(t)/25 + (7*sin(t))/25 - (t*cos(t))/5 - (t*sin(t))/10 + C6*exp(-2*t) 
>> pretty(ans)
     cos(t)   7 sin(t)   t cos(t)   t sin(t)
C5 - ------ + -------- - -------- - -------- + C6 exp(-2 t)
       25        25          5         10

7. y^IV-2y'''+2y''-2y'+y=5e^t

${\displaystyle y=\left(A+Bt\right)e^t+C\cos t+D\text{sen}t+\frac{5}{4}{t}^2{e}^t}$

>> y=dsolve('D4y-2*D3y+2*D2y-2*Dy+y=5*exp(t)')
y =(5*t^2*exp(t))/2 + C9*cos(t) + C7*exp(t) - C10*sin(t) + 
(5*exp(t)*cos(t)*(cos(t) + sin(t)))/4 - (5*t*exp(t)*(t + 2))/4 
- (5*exp(t)*sin(t)*(cos(t) - sin(t)))/4 + C8*t*exp(t)
>> y=simplify(y) 
y = (5*exp(t))/4 + (5*t^2*exp(t))/4 + C9*cos(t) + C7*exp(t) - C10*sin(t) -
 (5*t*exp(t))/2 + C8*t*exp(t)

8. y''+y=cos²(2t)+sen²(t/2)

${\displaystyle y=A\cos t+B\text{sen}t+1-\frac{1}{30}\cos 4t-\frac{1}{4}t\text{sen}t}$

>> y=dsolve('D2y+y=(cos(2*t))^2+(sin(t/2))^2')
y =sin(t)*(sin(3*t)/12 - sin(2*t)/8 - t/4 + sin(5*t)/20 + sin(t)) + 
C11*cos(t) - C12*sin(t) - cos(t)*(cos(2*t)/8 + cos(3*t)/12 - cos(5*t)/20 - 
cos(t) + 1/8)
>> y=simplify(y) 
y = C11*cos(t) - cos(t)/4 - cos(4*t)/30 - C12*sin(t) - (t*sin(t))/4 + 1

Ejercicios con condiciones iniciales

1. y''-5y'+6y=(12t-7)e^-t, y(0)=y'=0

>> y=dsolve('D2y-5*Dy+6*y=(12*t-7)*exp(-t)','y(0)=0','Dy(0)=0')
y =exp(2*t) - exp(3*t) - exp(-t)*(3*t - 1) + exp(-t)*(4*t - 1)
>> y=simplify(y) 
y =exp(-t)*(t + exp(3*t) - exp(4*t))

${\displaystyle y=e^{2t}-e^{3t}+t{e}^{-t}}$

2. y''-6y'+9y=t²-t+3, y(0)=4/3, y'(0)=1/27

>> y=dsolve('D2y-6*Dy+9*y=t^2-t+3','y(0)=4/3','Dy(0)=1/27')
y =t/27 + exp(3*t) - 3*t*exp(3*t) + t^2/9 + 1/3

${\displaystyle y=(1-3t)e^{3t}+\frac{t^2}{9}+\frac{t}{27}+\frac{1}{3}}$

3. y''+4y=4(sen2t+cos2t), y(π)=y'(π)=2π

>> dsolve('D2y+4*y=4*(sin(2*t)+cos(2*t))','y(pi)=2*pi','Dy(pi)=2*pi') 
ans =sin(2*t)/4 - cos(2*t)*(t - (2^(1/2)*sin(4*t + pi/4))/4) + 
sin(2*t)*(t - (2^(1/2)*cos(4*t + pi/4))/4) + cos(2*t)*(3*pi - 1/4)
>> simplify(ans)
ans =sin(2*t)/2 - t*cos(2*t) + t*sin(2*t) + 3*pi*cos(2*t)

4. y^IV-y=8e^t, y(0)=0, y'(0)=2, y''(0)=4, y'''(0)=6

>> y=dsolve('D4y-y=8*exp(t)','y(0)=0','Dy(0)=2','D2y(0)=4','D3y(0)=6')
y =2*exp(t) + 2*t*exp(t) - 2*exp(t)*sin(t)*(cos(t) + sin(t)) 
- 2*exp(t)*cos(t)*(cos(t) - sin(t))
 >> simplify(y)
ans =2*t*exp(t)

Ecuaciones de Euler

1. t²y''+ty'+y=t(6-lnt)

>> y=dsolve('t^2*D2y+t*Dy+y=t*(6-log(t))')
y =C14*cos(log(t)) - C13*sin(log(t)) + (t*cos(log(t))*(7*cos(log(t))
 - 6*sin(log(t)) - cos(log(t))*log(t) + sin(log(t))*log(t)))/2 
+ (t*sin(log(t))*(6*cos(log(t)) + 7*sin(log(t)) - cos(log(t))*log(t) - 
sin(log(t))*log(t)))/2
>> simplify(y)
ans = (7*t)/2 - (t*log(t))/2 + C14*cos(log(t)) - C13*sin(log(t))

2. t²y''-ty'-3y=-16ln(t)/t

>> y=dsolve('t^2*D2y-t*Dy-3*y=-16*log(t)/t')
y = (C15 + log(t) + 2*log(t)^2 + 1/4)/t + C16*t^3

3. t²y''-ty'+2y=tln(t)

>> y=dsolve('t^2*D2y-t*Dy+2*y=t*log(t)')
y =C18*t*cos(log(t)) - t*cos(log(t))*(sin(log(t)) - cos(log(t))*log(t)) 
+ t*sin(log(t))*(cos(log(t)) + sin(log(t))*log(t)) - C17*t*sin(log(t))
>> simplify(y)
 ans =t*(log(t) + C18*cos(log(t)) - C17*sin(log(t)))

4. t²y''-2ty'+2y=t²-2t+2

>> y=dsolve('t^2*D2y-2*t*Dy+2*y=t^2-2*t+2')
y =C20*t + C19*t^2 + t*(2*log(t) - t + 2/t) + t^2*(log(t) + (2*t - 1)/t^2)
>> simplify(y)
ans = 2*t + t^2*log(t) + C20*t + C19*t^2 + 2*t*log(t) - t^2 + 1

${\displaystyle y=A{t}^2+Bt+1+(2t+t^2)\ln t}$

C19-1=A; C20+2=B.

5. t²y''+4ty'+2y=2ln²t+12t

>> y=dsolve('t^2*D2y+4*t*Dy+2*y=2*(log(t))^2+12*t')
y = -(C22 + 3*t^2*log(t) + C21*t - t^2*log(t)^2 - (7*t^2)/2 - 2*t^3)/t^2
>> simplify(y)
ans =-(C22 + 3*t^2*log(t) + C21*t - t^2*log(t)^2 - (7*t^2)/2 - 2*t^3)/t^2
>> expand(ans)
ans =2*t - 3*log(t) + log(t)^2 - C21/t - C22/t^2 + 7/2

${\displaystyle y=\frac{A}{t^2}+\frac{B}{t}+\frac{7}{2}-3\ln t+{\ln }^{2}t+2t}$

6. t²y''+2ty''-4y'+4y/t=2t

>> y=dsolve('t^3*D3y+2*t^2*D2y-4*t*Dy+4*y=2*t^2')
 y = (t^2*log(t))/2 + C24*t + C23*t^2 + C25/t^2 - (5*t^2)/8

${\displaystyle y=At+B{t}^2+\frac{C}{t^2}+\frac{t^2\ln t}{2}}$

C23-5/8=B

7. t²y''-ty'-3y=5t³ln(t)

>> y=dsolve('t^2*D2y-t*Dy-3*y=5*t^3*log(t)')
 y =C27*t^3 + t^3*((5*log(t)^2)/8 - (5*log(t))/16 + C26/(4*t^4))

Sistemas de ecuaciones diferenciales

1. ${\displaystyle \begin{array}{l}\frac{dx}{dt}=-7x+y+5\\ \frac{dy}{dt}=-2x-5y-37t\end{array}\}x(0)=y(0)=0}$

>> syms  x y;
>> [x, y]=dsolve('Dx=-7*x+y+5','Dy=-2*x-5*y-37*t','x(0)=0','y(0)=0') 
x =1 - exp(-6*t)*cos(t) - t
 y = exp(-6*t)*sin(t) - exp(-6*t)*cos(t) - 7*t + 1

x=1-t-e^-6tcost, y=1-7t-e^-6tcost+e^-6tsent

2. ${\displaystyle \begin{array}{l}\frac{dx}{dt}=3x-\frac{1}{2}y-3t^2-\frac{1}{2}t+\frac{3}{2}\\ \frac{dy}{dt}=2y-2t-1\end{array}\}}$

>> [x y]=dsolve('Dx=3*x-y/2-3*t^2-t/2+3/2','Dy=2*y-2*t-1')
x = (exp(2*t)*(C30 + exp(-2*t)*(t + 1)))/2 + exp(3*t)*
(C31 + (exp(-3*t)*(2*t - 1)*(t + 1))/2)
y =exp(2*t)*(C30 + exp(-2*t)*(t + 1))
>> simplify([x y])
ans = [ t + t^2 + (C30*exp(2*t))/2 + C31*exp(3*t), t + C30*exp(2*t) + 1]

3. ${\displaystyle \begin{array}{l}\frac{dx}{dt}+2\frac{dy}{dt}=17x+8y\\ 13\frac{dx}{dt}=53x+2y\end{array}\}x(0)=2,y(0)=-1}$

>> [x y]=dsolve('Dx+2*Dy=17*x+8*y','13*Dx=53*x+2*y','x(0)=2','y(0)=-1') 
x =exp(3*t) + exp(5*t)
y =6*exp(5*t) - 7*exp(3*t)

4. ${\displaystyle \begin{array}{l}\frac{dx}{dt}+\frac{dy}{dt}=e^{-t}-y\\ 2\frac{dx}{dt}+\frac{dy}{dt}=\text{sen}t-2y\end{array}\}x(0)=-2,y(0)=1}$

>> [x y]=dsolve('Dx+Dy=exp(-t)-y','2*Dx+Dy=sin(t)-2*y','x(0)=-2','y(0)=1')
x =- 2*t - exp(-t) - 2^(1/2)*cos(t - pi/4)
y =cos(t) - 2*exp(-t) + 2
 >> x=expand(x)
 x = - 2*t - exp(-t) - cos(t) - sin(t)