Consider the first order differential equation $y^{\text{'}}+(\frac{x}{x^2-16})y=\frac{e^x}{x-6}.$

For each $of$ the initial conditions below, determine the largest interval $y(-7)=-3\mathrm{.}14=y(-0\mathrm{.}5)=1\mathrm{.}7=y(0)=0=y(5\mathrm{.}5)=-4=y(8)=-5\mathrm{.}5=aon$

which the existence and uniqueness theorem for first order linear differential equations

guarantees the existence $of$ a unique solution.

$(a)y(-7)=-3\mathrm{.}14=>$ Interval $=$

$(b)y(-0\mathrm{.}5)=1\mathrm{.}7=>$ Interval $=$

$(c)y(0)=0=>$ Interval $=$

$(d)y(5\mathrm{.}5)=-4=>$ Interval $=$

$(e)y(8)=-5\mathrm{.}5=>$ Interval $=$