Solve the given equation by using Laplace transforms: \( \begin{array}{l}6.1 \quad y^{\prime}(t)+4 y(t)=e^{-4 t} \text { if } y(0)=2 \\ 6.2 \quad y^{\prime \prime}(t)-7 y^{\prime}(t)+6 y=\delta(t-2) \text { if } y(0)=0 \text { and } y^{\prime}(0)=0\end{array} \)

4. Which theme does this passage want to express to the reader a. good overcomes evill b. honesty is the best policy c. the importance of family d. doos are fiendlier thon cots

3. (Unisc-SC) A parte real do número complexo \( z=\frac{1+(3 i)^{2}}{1-i} \) é \( \begin{array}{ll}\text { i) } 1 & \text { d) }-2 \\ \text { b) }-1 & \text { e) }-4\end{array} \)

State any two commonly used software tools for Time Series Analysis. (2Marks) Give an expression for \( X_{t}-5 X_{t-1}+7 X_{t-2}-3 X_{t-3} \quad \) in terms of second order di fferences.

\( \begin{array}{ll}2.1 .2 & \frac{\sqrt{2}}{\sqrt{2}+1}+\frac{4}{\sqrt{2}} \\ 2.2 & \text { Solve for } x \\ 2.2 .1 & \left(\frac{1}{2}\right)^{x}=32 \\ 2.2 .2 & \sqrt{(x-2)^{-3}}=64\end{array} \)

\( y ^ { \prime } - \frac { y } { x } = 1,5 \sin x \)