Consider the following initial value problem.

$y^{\text{'}}=y-t,y(0)=2.$

Using a calculator, compute an approximation to $y(2)$ using the following methods. How close is each to the exact solution? Round as little as possible between steps.

$($a$)$ Compute the exact solution $($no calculator necessary$).$

Solve the differential equation using the method of integrating factors. Then plug

$int=2$

$y(2)=2$

$($b$)$ Forward Euler with step size $t=0\mathrm{.}5$

Some intermediate steps:

$y_1=1,$

Error in $y(2)$:$0\mathrm{.}46875$

$($c$)$ Forward Euler with step size $t=0\mathrm{.}25$

Some intermediate steps:

$y_1=1\mathrm{.}5,y_3$~~$0\mathrm{.}918,y_5$~~$0\mathrm{.}872,y_7$~~$1\mathrm{.}346$

Error in $y(2)$:$20\mathrm{.}22497$

$($d$)$ Backward Euler with step size $t=0\mathrm{.}5$

Some intermediate steps:

$y_1$~~$1\mathrm{.}417,y_3$~~$1\mathrm{.}602$

Error in $y(2)$:$0\mathrm{.}4012$

$($e$)$ Backward Euler with step size $t=0\mathrm{.}25$

Some intermediate steps:

$y_1=1\mathrm{.}6125,y_3$~~$1\mathrm{.}1845,y_5$~~$1\mathrm{.}2306,y_7$~~$1\mathrm{.}7601$

Error in $y(2)$:$0\mathrm{.}20806$Consider the following initial value problem.

$y^{\text{'}}=y-t,y(0)=2.$

Using a calculator, compute an approximation to $y(2)$ using the following methods. How close is each to the exact solution? Round as little as possible between steps.

$($a$)$ Compute the exact solution $($no calculator necessary$).$

Solve the differential equation using the method of integrating factors. Then plug

$int=2$

$y(2)=2$

$($b$)$ Forward Euler with step size $t=0\mathrm{.}5$

Some intermediate steps:

$y_1=1,$

Error in $y(2)$:$0\mathrm{.}46875$

$($c$)$ Forward Euler with step size $t=0\mathrm{.}25$

Some intermediate steps:

$y_1=1\mathrm{.}5,y_3$~~$0\mathrm{.}918,y_5$~~$0\mathrm{.}872,y_7$~~$1\mathrm{.}346$

Error in $y(2)$:$20\mathrm{.}22497$

$($d$)$ Backward Euler with step size $t=0\mathrm{.}5$

Some intermediate steps:

$y_1$~~$1\mathrm{.}417,y_3$~~$1\mathrm{.}602$

Error in $y(2)$:$0\mathrm{.}4012$

$($e$)$ Backward Euler with step size $t=0\mathrm{.}25$

Some intermediate steps:

$y_1=1\mathrm{.}6125,y_3$~~$1\mathrm{.}1845,y_5$~~$1\mathrm{.}2306,y_7$~~$1\mathrm{.}7601$

Error in $y(2)$:$0\mathrm{.}20806$