$(a)$ Find $by$ guessing a one$-$parameter family $of$ solutions $of$ the differ$-$

ential equation $x{y}^{\text{'}}=y.$ Verify that each member $of$ the family $isa$

solution $of$ the initial value problem $x{y}^{\text{'}}=y,y(0)=0.$

$(b)$ Explain $(a)by$ determining a region $Rin$ the $xy-$plane for which

the differential equation $x{y}^{\text{'}}=y$ would have a unique solution through

the point $(x_0,y_0)inR.$

$(c)$ Verify that the piecewise defined function

$y=\{\begin{array}{cc}0,& x<mn>0</mn>\\ x,& x0\end{array}$

satisfies the condition $y(0)=0.$ Determine whether the function $is$ also

a solution $of$ the IVP $in(a).$