Question

find the genaral solution of the given system of equations
dx/dt=-x+y
dy/dt=x+6y+y
dz/dt=7y-z

Answer 1

Call x = y₁, y = y₂, and z = y₃.

express the system as Y' = A . Y

       [ y1' ]           [ x ]    [ y1 ]
Y' = [ y2' ]     Y = [ y ] = [ y2 ]
       [ y3' ]           [ z ]    [ y3 ]

y₁' = -y₁ + y₂
y₂' = y₁ + 6 y₂ + y₃   or   y₁ + 7 y₂   which one is right ?
y₃'  = 7 y₂ - y₃

        -1  1   0          -1  1   0
A =   1   6   1    or   1  7   0
        0   7   -1         0  7   -1

find Eigen values λi of A.   Write   Deteminant | λ I - A | = 0
From the cubic polynomial equation in λ, find its values. They are the Eigen values.


Find Eigen vectors Vi of A.
       write   A X = λ X         Then solve for x1, x2, x3 in terms of one another.
        Find  Vi = V₁, V₂ and V₃ corresponding to λ₁  λ₂ and λ₃.
              Vi = transpose of [ 1 x2/x1  x3/x1 ]

        This is to be done for each Eigen value λi.

Then answer is :

       Y' = c₁ V₁ e^(λ₁t) + c₂ V₂ e^(λ₂ t) + c₃ V₃ e^(λ₃ t)
     
   So y₁ = x = addition of the first elements of V1, V2 and V3 after multiplying with the corresponding coefficients.
  similarly others.