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Find the genaral solution of the given system of equations

dx/dt=-x+y

dy/dt=x+6y+y

dz/dt=7y-z

1
by mrambabu1600

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

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dx/dt=-x+y

dy/dt=x+6y+y

dz/dt=7y-z

by mrambabu1600

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

Log in to add a comment

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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.

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.

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.