Consider the differontinl equation $\frac{dy}{dx}=y^2(2x+2).$ Lot $y=f(x)$ be the particulur solusion to tho difforential equation with intiat condition $f(0)=-1.$

a$)$ Find the equation of a saggent litac at $x=0.(1$ pt$)$

b$)$ Use Euler's method, starting with $x=0$ with four steps of equal size, to approximats $)$

c$)$ Find $y=f(x),$ the particular solution to the differential equation with initial condition $f(0)=-1.($You must show your steps using separation of variables.$)(4$ pts$)$

d$)$ Consider another differential equation: $\frac{dP}{dt}=kP(1-\frac{P}{L})$ which has a solution form of $P=\frac{L}{1+b{e}^{-kt}}.$ Given that $(0,20)$ and $(3,50)$ are the specific solutions for the function $P,$ and given that $\underset{t}{lim}P(t)=100,$ find the values of $L,b,$ and $k.$ Y ou can leave your ans for $k$ either with LN or round to three decimal places. Please show your work or reasoning for each. $(3$ pts$)$