\( \left\{ \begin{array} { l } { y ^ { \prime } = x [ 3 - 2 y ] } \\ { y ( 0 ) = 0 } \end{array} \right. \)

To tackle the initial value problem given by the equation \( y' = x[3 - 2y] \) with \( y(0) = 0 \), we can employ the method of separation of variables. Rearranging the equation yields \(\frac{dy}{3 - 2y} = x \, dx\). Once separated, we can integrate both sides to find the function \( y(x) \) that satisfies the given condition. When integrating, remember to account for solving the logarithmic forms and applying the initial condition \( y(0) = 0 \). This will allow you to determine any constants arising from the integration, leading you to a specific solution. Solving ODEs often requires attention to detail in both integration and application of initial values. A common mistake is forgetting to check if exponentiating back from logarithms can introduce extraneous solutions or not satisfying the initial conditions. So pay close attention to the mathematics, and verify your solutions!