Question

In each of Problems 1-4 find the general solution of the given system of differential equations.

2. x˙= ( 1 −5 0)
( 1 −3 0)x
( 0 0 1)

Answer 1

The general solution of the system of differential equations is x(t) = c₁e^t[5, 3, 1] + c₂e^(-3t)[0, 0, 1], where c₁ and c₂ are arbitrary constants.

The given system of differential equations, represented as x =
`\left[\begin{array}{ccc}1 & -5 & 0\\1 & -3 & 0\\0 & 0 & 1\end{array}\right]` x, is a linear system. To find the general solution, we first determine the eigenvalues of the coefficient matrix. The eigenvalues are 1 and -3 with corresponding eigenvectors [5, 3, 1] and [0, 0, 1].

The general solution is then expressed as a linear combination of these eigenvectors multiplied by their respective exponential terms, where c_1​ and c_2 are arbitrary constants. The solution accounts for the system's behavior over time, considering the influence of each eigenvector and the associated exponential term.