$1.$ You will be writing the code the have MATLAB perform Variation of Paramters to solve the differential

equation

t$3$y$+$ t$2$y$2$ty$+2$y $=2$t$4,$ t $>0,$

where y$1($t$)=$ t$,$ y$2($t$)=$ t$2,$ y$3($t$)=1$

t are solutions to the corresponding homogeneous differential

equation.

$($a$)$ Define the variables u$,$ v$,$w$,$ t$,$ c$1,$ c$2,$ c$3$ in MATLAB. Then use the above differential equation and

solutions to define y$1,$ y$2,$ y$3,$ f in MATLAB. $($You do not need to define these as functions as they

will not be evaluated at any values.$)$

$($b$)$ Use your results from part $($a$)$ to define the system of equations necessary to perform variation of

parameters. $($Use u$,$ v$,$w in place of u$$

$1,$ u$$

$2,$ u$$

$3,$ respectively.$)$ Call this system $$eqns$.$ This should

only be one line of code as you will tell MATLAB to perform any necessay derivatives within the

system itself.

$($c$)$ Solve the system $$eqns$$ for u$,$ v$,$ and w$.($Explicitly use $[$u v w$]$ in your code.$)$

$($d$)$ Find the general solution y of the differential equation. This should only be one line of code as

you will tell MATLAB to perform any necessay integrals. $($MATLAB does not include a constant

of integration for indefinite integrals. But your solution still needs to include them.$)$