Gate 2016 EE -Set 2 - Question 49

Given, \dot{x}=Ax

With initial condition x(0) at t = 0 and x(0) = \alpha

Eigen values are \lambda _{1} and \lambda _{2}

\phi (t)=\begin{bmatrix}e^{\lambda _{1}t} &0\\0&e^{\lambda _{2}t} \end{bmatrix}

Response due to initial condition \Rightarrow x(t)=\phi (t)\cdot x(0)

x(t)=\begin{bmatrix}e^{\lambda _{1}t} &0\\0&e^{\lambda _{2}t} \end{bmatrix}\begin{bmatrix}\alpha\\0\end{bmatrix}

x(t)=\alpha e^{\lambda _{1}t}