1. **State the problem:** Solve the initial-value problem $$\frac{dy}{dt} = \frac{4y}{t}, \quad y(2) = 3.$$\n\n2. **Identify the type of differential equation:** This is a first-order separable differential equation. We can separate variables to solve it.\n\n3. **Separate variables:** Rewrite as $$\frac{dy}{y} = \frac{4}{t} dt.$$\n\n4. **Integrate both sides:** $$\int \frac{1}{y} dy = \int \frac{4}{t} dt.$$\nThis gives $$\ln|y| = 4 \ln|t| + C,$$ where $C$ is the constant of integration.\n\n5. **Simplify the right side:** Using logarithm properties, $$\ln|y| = \ln|t|^4 + C.$$\n\n6. **Exponentiate both sides to solve for $y$:** $$y = e^{\ln|t|^4 + C} = e^C \cdot |t|^4.$$\nLet $A = e^C$, so $$y = A t^4.$$\n\n7. **Use the initial condition to find $A$:** Given $y(2) = 3$, substitute $t=2$:\n$$3 = A \cdot 2^4 = A \cdot 16,$$\nso $$A = \frac{3}{16}.$$\n\n8. **Final solution:** $$\boxed{y = \frac{3}{16} t^4}.$$