$1.$ x$($dy$)/($dx$)-4$y$=$x$^(5)$e$^(\text{'})$

$2.$ y$^(2)($dy$)/($dx$)-$xy$+$x$^(3)=0$

$3.($dy$)/($dx$)=4($y$^(2)+1),$y$((\backslash$pi $)/(4))=1$

$4.($dy$)/($dx$)=$xe$^(4$x$+5$y$)$

Answer the following questions about the equations.

$1.$

a$-$ Which one of the differential equations is separable and NOT an initial value problem?

b$-$ Write your answer from part $($a$)$ in the form P$($x$)$ dx $=$ Q$($y$)$ dy for some functions P and Q$.$ Show your work.

c$-$ Solve the differential equation. Show your work.

$2.$

a$-$ Which one of the differential equations is separable and IS an initial value problem?

b$-$ Write your answer from part $($a$)$ in the form P$($x$)$ dx $=$ Q$($y$)$ dy for some functions P and Q$.$ Show your work.

c$-$ Solve the initial value problem. Show your work.

$3.$

a$-$ Is the differential equation you chose in problem $1$ also linear? Write a $1-3$ sentence explanation of your answer.

b$-$ Now ignore the differential equations you chose from problem $1$ and problem $2.$ Of the remaining two differential equations, which one is linear?

c$-$ Solve the linear differential equation you chose in part $($b$).$ Show your work.

$4.$

a$-$ There should be one differential equation left you have not chosen. Which one is left?

b$-$ Use a computer algebra system$/$online graphing utility to sketch the direction field of this differential equation on the window $10$ x $10,10$ y $10.$ On your direction field, sketch the solution satisfying the initial condition $(0,2).$ One graphing utility is Desmos $($Direction$/$Slope Fields$).$ For help with Desmos, click on the Help icon $(?)$ and watch the video tutorials.