1. **State the problem:** Solve the differential equation $$x^2 y'' - 4x y' + 6y = 4x - 6$$ using the method of variation of parameters or a similar method. 2. **Identify the type of equation:** This is a Cauchy-Euler (or equidimensional) differential equation. The associated homogeneous equation is $$x^2 y'' - 4x y' + 6y = 0$$. 3. **Solve the homogeneous equation:** Assume a solution of the form $$y = x^m$$. Substitute into the homogeneous equation: $$x^2 (m(m-1)x^{m-2}) - 4x (m x^{m-1}) + 6 x^m = 0$$ Simplify: $$m(m-1) x^m - 4m x^m + 6 x^m = 0$$ $$x^m [m(m-1) - 4m + 6] = 0$$ $$m^2 - m - 4m + 6 = 0$$ $$m^2 - 5m + 6 = 0$$ 4. **Solve the characteristic equation:** $$m^2 - 5m + 6 = 0$$ Factor: $$(m-2)(m-3) = 0$$ So, $$m = 2$$ or $$m = 3$$. 5. **General solution to homogeneous equation:** $$y_h = C_1 x^2 + C_2 x^3$$ 6. **Find a particular solution $$y_p$$:** Use variation of parameters or method of undetermined coefficients. Since the right side is $$4x - 6$$, try a particular solution of the form: $$y_p = A x + B$$ 7. **Compute derivatives:** $$y_p' = A$$ $$y_p'' = 0$$ 8. **Substitute into original equation:** $$x^2 (0) - 4x (A) + 6 (A x + B) = 4x - 6$$ Simplify: $$-4 A x + 6 A x + 6 B = 4 x - 6$$ $$ (2 A) x + 6 B = 4 x - 6$$ 9. **Equate coefficients:** For $$x$$ terms: $$2 A = 4 \\ A = 2$$ For constants: $$6 B = -6 \\ B = -1$$ 10. **Particular solution:** $$y_p = 2 x - 1$$ 11. **General solution:** $$y = y_h + y_p = C_1 x^2 + C_2 x^3 + 2 x - 1$$