# 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 $$\Rightarrow x(t)=\phi (t)\cdot x(0)$$

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

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