Question $2$

Solve the following initial value problems, and determine the intervals through which the solutions are valid:

$(a)y^{\text{'}}=\frac{x{y}^3}{\sqrt[\phantom{2}]{1+x^2}},y(0)=-1$

$(b)\frac{dy}{dx}=y^{2}sint,y(0)=\frac{1}{2}$

$(c)y^{\text{'}}=\frac{3x^2+4x-4}{\sqrt[\phantom{2}]{2y-4}},y(1)=3$