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Tolnaio

Answered question

2021-01-23

The coefficient matrix for a system of linear differential equations of the form

y^{1} = A y

has the given eigenvalues and eigenspace bases. Find the general solution for the system

λ_{1} = 2 i \Rightarrow {[\begin{matrix} 1 + i \\ 2 - i \end{matrix}]}, λ_{2} = - 2 i \Rightarrow {[\begin{matrix} 1 - i \\ 2 + i \end{matrix}]}

dessinemoie

Skilled2021-01-24Added 90 answers

By theorem 6.19 we know that the solution is

y = c_{1} e^{λ_{1} t} u_{1} + \dots + c_{n} e^{λ_{n} t} u_{n}

with

λ_{i}

the eigenvalues of the matrix A nad

u_{i}

the eigenvectors Thus for this case we then obtain the general solution:

[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = y = c_{1} e^{2 i t} [\begin{matrix} 1 + i \\ 2 - i \end{matrix}] + c_{2} e^{- 2 i t} [\begin{matrix} 1 - i \\ 2 + i \end{matrix}]

Thus we obtain:

y_{1} = c_{1} (\cos (2 t) - \sin (2 t)) + c_{2} (\cos (2 t) + \sin (2 t))

y_{2} = c_{1} (2 \cos (2 t) + \sin (2 t)) + c_{2} (\cos (2 t) - 2 \sin (2 t))

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