b) \( \frac{d y}{d x}=-\frac{3 y}{2 x} ; y(2)=4 \)

To solve the differential equation \(\frac{d y}{d x} = -\frac{3 y}{2 x}\) with the initial condition \(y(2) = 4\), we can use the method of separation of variables. Rearranging gives us \(\frac{dy}{y} = -\frac{3}{2} \frac{dx}{x}\). Integrating both sides, we obtain \(\ln |y| = -\frac{3}{2} \ln |x| + C\). Applying the exponential function, we have \(y = K x^{-\frac{3}{2}}\), where \(K = e^C\). Using the initial condition, \(4 = K \cdot 2^{-\frac{3}{2}}\) allows us to solve for \(K = 4 \cdot 2^{\frac{3}{2}} = 4 \cdot \sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2}\). Thus, the solution is \(y = 8\sqrt{2} \cdot x^{-\frac{3}{2}}\). To verify, substitute \(x = 2\): \(y = 8\sqrt{2} \cdot (2^{-\frac{3}{2}}) = 8\sqrt{2} \cdot \frac{1}{2\sqrt{2}} = 4\). The solution checks out!