Równania liniowe

1847 days ago by Henryk

A = matrix(SR,[[1,-2],[5,1]]) # to jest macierz z zadania 4 t = var('t') # szukamy rozwiązania x(y),y(t) x = function('x',t) y = function('y',t) C_1 = var('C_1') C_2 = var('C_2') S_1 = A.right_eigenmatrix()[1] S = matrix(SR,[[-I*sqrt(2/5),I*sqrt(2/5)],[1,1]]) # S to jest macierz wektorów własnych S_1; S; A 
       
[              1               1]
[ 1/2*I*sqrt(10) -1/2*I*sqrt(10)]
[-I*sqrt(2/5)  I*sqrt(2/5)]
[           1            1]
[ 1 -2]
[ 5  1]
S_inverse = S.inverse(); S_inverse 
       
[ 5/4*I*sqrt(2/5)              1/2]
[-5/4*I*sqrt(2/5)              1/2]
S*S_inverse 
       
[1 0]
[0 1]
D = S_inverse*A*S; D; matrix(CDF,D) 
       
[ 1/4*I*(-5*I*sqrt(2/5) - 10)*sqrt(2/5) - 5/2*I*sqrt(2/5) + 1/2
-1/4*I*(-5*I*sqrt(2/5) - 10)*sqrt(2/5) - 5/2*I*sqrt(2/5) + 1/2]
[  1/4*I*(5*I*sqrt(2/5) - 10)*sqrt(2/5) + 5/2*I*sqrt(2/5) + 1/2 
-1/4*I*(5*I*sqrt(2/5) - 10)*sqrt(2/5) + 5/2*I*sqrt(2/5) + 1/2]
[1.0 - 3.16227766017*I                     0]
[                    0 1.0 + 3.16227766017*I]
matrix(CDF,S*D*S_inverse) 
       
[ 1.0 -2.0]
[ 5.0  1.0]
D_1 = matrix(SR,[[exp(t*(1-I*sqrt(10))),0],[0,exp(t*(1+I*sqrt(10)))]]); D_1; (matrix(SR,[[C_1,C_2]])*S*D_1*S_inverse) 
       
[e^((-I*sqrt(10) + 1)*t)                       0]
[                      0  e^((I*sqrt(10) + 1)*t)]
[5/4*I*(-I*sqrt(2/5)*C_1 + C_2)*sqrt(2/5)*e^((-I*sqrt(10) + 1)*t) -
5/4*I*(I*sqrt(2/5)*C_1 + C_2)*sqrt(2/5)*e^((I*sqrt(10) + 1)*t)      
1/2*(-I*sqrt(2/5)*C_1 + C_2)*e^((-I*sqrt(10) + 1)*t) +
1/2*(I*sqrt(2/5)*C_1 + C_2)*e^((I*sqrt(10) + 1)*t)]
x = 5/4*I*(-I*sqrt(2/5)*C_1 + C_2)*sqrt(2/5)*e^((-I*sqrt(10) + 1)*t) - 5/4*I*(I*sqrt(2/5)*C_1 + C_2)*sqrt(2/5)*e^((I*sqrt(10) + 1)*t) y = 1/2*(-I*sqrt(2/5)*C_1 + C_2)*e^((-I*sqrt(10) + 1)*t) + 1/2*(I*sqrt(2/5)*C_1 + C_2)*e^((I*sqrt(10) + 1)*t) 
       
print (derivative(x,t)-x-5*y).simplify() print (derivative(y,t)+2*x-y).simplify() 
       
1/20*(-I*sqrt(10) + 1)*(sqrt(2)*sqrt(5)*C_1 +
5*I*C_2)*sqrt(2)*sqrt(5)*e^((-I*sqrt(10) + 1)*t) + 1/20*(I*sqrt(10)
+ 1)*(sqrt(2)*sqrt(5)*C_1 - 5*I*C_2)*sqrt(2)*sqrt(5)*e^((I*sqrt(10)
+ 1)*t) - 1/20*(sqrt(2)*sqrt(5)*C_1 -
5*I*C_2)*sqrt(2)*sqrt(5)*e^((I*sqrt(10) + 1)*t) -
1/20*(sqrt(2)*sqrt(5)*C_1 + 5*I*C_2)*sqrt(2)*sqrt(5)*e^((-I*sqrt(10)
+ 1)*t) - 1/2*(-I*sqrt(2)*sqrt(5)*C_1 + 5*C_2)*e^((-I*sqrt(10) +
1)*t) - 1/2*(I*sqrt(2)*sqrt(5)*C_1 + 5*C_2)*e^((I*sqrt(10) + 1)*t)
1/10*(-I*sqrt(10) + 1)*(-I*sqrt(2)*sqrt(5)*C_1 +
5*C_2)*e^((-I*sqrt(10) + 1)*t) + 1/10*(I*sqrt(10) +
1)*(I*sqrt(2)*sqrt(5)*C_1 + 5*C_2)*e^((I*sqrt(10) + 1)*t) +
1/10*(sqrt(2)*sqrt(5)*C_1 - 5*I*C_2)*sqrt(2)*sqrt(5)*e^((I*sqrt(10)
+ 1)*t) + 1/10*(sqrt(2)*sqrt(5)*C_1 +
5*I*C_2)*sqrt(2)*sqrt(5)*e^((-I*sqrt(10) + 1)*t) -
1/10*(-I*sqrt(2)*sqrt(5)*C_1 + 5*C_2)*e^((-I*sqrt(10) + 1)*t) -
1/10*(I*sqrt(2)*sqrt(5)*C_1 + 5*C_2)*e^((I*sqrt(10) + 1)*t)
x = sqrt(5/2)*C_2*e^(t)*sin(sqrt(10)*t)+C_1*e^(t)*cos(sqrt(10)*t) y = C_2*e^(t)*cos(sqrt(10)*t)-sqrt(2/5)*C_1*e^(t)*sin(sqrt(10)*t) x; y 
       
sqrt(5/2)*C_2*e^t*sin(sqrt(10)*t) + C_1*e^t*cos(sqrt(10)*t)
-sqrt(2/5)*C_1*e^t*sin(sqrt(10)*t) + C_2*e^t*cos(sqrt(10)*t)
print (derivative(x,t)-x-5*y).simplify() print (derivative(y,t)+2*x-y).simplify() 
       
1/2*sqrt(2)*sqrt(5)*sqrt(10)*C_2*e^t*cos(sqrt(10)*t) +
sqrt(2)*sqrt(5)*C_1*e^t*sin(sqrt(10)*t) -
sqrt(10)*C_1*e^t*sin(sqrt(10)*t) - 5*C_2*e^t*cos(sqrt(10)*t)
-1/5*sqrt(2)*sqrt(5)*sqrt(10)*C_1*e^t*cos(sqrt(10)*t) +
sqrt(2)*sqrt(5)*C_2*e^t*sin(sqrt(10)*t) -
sqrt(10)*C_2*e^t*sin(sqrt(10)*t) + 2*C_1*e^t*cos(sqrt(10)*t)