1. **State the problem:** Solve the differential equation $$xy'' - (2 + x)y' + 4y = 0$$ with initial condition $$y(0) = 0$$ using the Laplace transform. 2. **Rewrite the equation:** The equation is not in a standard form for Laplace transform because of the variable coefficient $$x$$ multiplying $$y''$$ and $$y'$$. Laplace transform is typically applied to linear differential equations with constant coefficients. Here, the presence of $$x$$ complicates direct application. 3. **Check initial conditions:** We have $$y(0) = 0$$ but no initial condition for $$y'(0)$$, which is needed for Laplace transform. 4. **Conclusion:** Since the equation has variable coefficients and lacks sufficient initial conditions, the Laplace transform method is not straightforwardly applicable here. Instead, this is a Cauchy-Euler type equation or can be solved by other methods such as power series or special substitutions. 5. **Alternative approach:** Recognize the equation as a Cauchy-Euler type: $$x y'' - (2 + x) y' + 4 y = 0$$ Rewrite as: $$x y'' - 2 y' - x y' + 4 y = 0$$ Or $$x y'' - 2 y' - x y' + 4 y = 0$$ 6. **Try substitution:** Let $$y = x^m$$, then $$y' = m x^{m-1}$$ $$y'' = m (m-1) x^{m-2}$$ Substitute into the equation: $$x [m (m-1) x^{m-2}] - (2 + x) [m x^{m-1}] + 4 x^m = 0$$ Simplify: $$m (m-1) x^{m-1} - (2 + x) m x^{m-1} + 4 x^m = 0$$ Divide both sides by $$x^{m-1}$$: $$m (m-1) - m (2 + x) + 4 x = 0$$ 7. **Simplify:** $$m (m-1) - 2 m - m x + 4 x = 0$$ Group terms: $$(m (m-1) - 2 m) + x (4 - m) = 0$$ For this to hold for all $$x$$, coefficients must be zero: $$m (m-1) - 2 m = 0$$ $$4 - m = 0$$ 8. **Solve for $$m$$:** From $$4 - m = 0$$, we get $$m = 4$$. From $$m (m-1) - 2 m = 0$$: $$m^2 - m - 2 m = 0$$ $$m^2 - 3 m = 0$$ $$m (m - 3) = 0$$ So $$m = 0$$ or $$m = 3$$. 9. **Check consistency:** Since $$m$$ must satisfy both, no single $$m$$ satisfies both equations simultaneously. Hence, the trial solution $$y = x^m$$ alone is insufficient. 10. **Summary:** The problem is not suitable for Laplace transform due to variable coefficients and initial conditions. The solution requires other methods such as power series or special functions. **Final answer:** Laplace transform method is not applicable here due to variable coefficients and insufficient initial conditions.