Consider the ODE

$$

x

$$

$4$x

$$

$5$x $2$t ICs x$0$

$1,,$ x

$$

$0$

$2$

$1)$ Solve x$($t$)$ using Laplace transformation technique. Time constant? How fast do the

effects of ICs disappear? Plot the solution, assess the frequency.

$2)$ Solve x$($t$)$ numerically $($SIMULINK only$).$ Plot x$($t$)$ and display that they are the same as in

$(1)$ above.

$3)$ Simply changing the ICs $($choose arbitrarily$)$ plot the corresponding x$($t$)$s for these

different ICs. Show that the claim of predicted $$disappearing act$$ of the ICs in $(1)$

holds..

$4)$ Challenge problem: How do you solve for

x

$($t$)$

numerically $($using SIMULINK and

Laplace$)!!$ This is a little tricky. Hint:

$0$

Laplace $($x

$($t$))$ s $*$ X $($s$)$ x

$!!$ Also

think how we create the IC block in SIMULINK.... Prove that your solution matches that

x

$($t$)$

you can obtain using only Laplace as in $(1)...$