# Find the genaral solution of the given system of equationsdx/dt=-x+ydy/dt=x+6y+ydz/dt=7y-z

1
by mrambabu1600
dy/dt = x+6y + z or y? in the second line

2014-10-22T03:53:48+05:30

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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.