A calculus problem by Rocco Dalto

Calculus Level pending

\[\begin{cases} \dfrac{dx}{dt} = 3x + y - 2z \\ \dfrac{dy}{dt} = x + y \\ \dfrac{dz}{dt} = x + z \end{cases} \]

The general solution to the above differential system is as follows:

\[\begin{cases} x(t) = c_{1}x_{1}(t) + c_{2}x_{2}(t) + c_{3}x_{3}(t) \\ y(t) = c_{1}y_{1}(t) + c_{2} y_{2}(t) + c_{3}y_{3}(t) \\ z(t) = c_{1}z_{1}(t) + c_{2}z_{2}(t) + c_{3}z_{3}(t) \end{cases} \] where \[ \begin{vmatrix}{x_{j}(t)} \\{y_{j}(t)} \\ {z_{j}(t)} \\ \end{vmatrix} =
\begin{vmatrix}{A_{j}} \\{B_{j}} \\ {C_{j}} \\ \end{vmatrix} * e^{\lambda_{j} t} \]

for \( (1 <= j <= 2) \), and \[ \begin{vmatrix}{x_{3}(t)} \\{y_{3}(t)} \\ {z_{3}(t)} \\ \end{vmatrix} =
\begin{vmatrix}{A_{3} + A_{2} * t} \\{B_{3} + B_{2} * t}\\ {C_{3} + C_{2} * t} \\ \end{vmatrix} * e^{\lambda_{2}t } \]

\(\)

If \( C_{1} = 1, \: C_{2} = 1, \: A_{3} = 2 \) and \(x(0) = 0\), \(y(0) = 2\), and \(z(0) = 3 \). Find \( c_{1} + c_{2} + c_{3} \).

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