(Solved): To solve the separable differential equation \[ \frac{x^{2}}{y^{2}-3} \frac{d y}{d x}=\frac{1}{2 y ...

To solve the separable differential equation \[ \frac{x^{2}}{y^{2}-3} \frac{d y}{d x}=\frac{1}{2 y} \] we must find two separate integrals: \[ \begin{array}{ll} \int & d y= \\ \text { and } & d x= \end{array} \] The first integral we integrate by substitution: \( \begin{array}{ll}u= & \text { help (formulas) } \\ d u= & \text { help (formulas) }\end{array} \) Solving for \( y \) we get one positive solution \[ y= \] help (formulas) And one negative solution \[ y= \] help (formulas) Note: You must simplify all arbitrary constants down to one constant \( k \). Find the particular solution satisfying the initial condition \[ y(1)=-8 \] \( y(x)= \) help (formulas)