Solving First$-$Order Differential Equations

Consider the following first$-$order differential equations:

$x\frac{dy}{dx}-4y=x^5{e}^x$

$y^2\frac{dy}{dx}-xy+x^3=0$

$\frac{dy}{dx}=4(y^2+1),y(\frac{}{4})=1$

$\frac{dy}{dx}=x{e}^{4x+5y}$

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 $ISan$ 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 $a1-3$ 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?

$(b)$ 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 syste$\frac{m}{o}$nline graphing utility $to$ sketch the direction field $of$ this differential equation $on$ the window $-10x10,-10y10.On$ your direction field, sketch the solution satisfying the initial condition $(0,-2).$ One graphing utility $is$ Desmos

$(Directio\frac{n}{S}$lope Fields$).$