1. **State the problem:** Solve the differential equation $$\frac{d^2x}{dt^2} - 2\frac{dx}{dt} = e^t (t - 3)$$ with initial conditions $$x(0) = 2$$ and $$x'(0) = 0$$. 2. **Identify the type of equation:** This is a nonhomogeneous linear second-order differential equation with constant coefficients. 3. **Solve the homogeneous equation:** $$\frac{d^2x}{dt^2} - 2\frac{dx}{dt} = 0$$ The characteristic equation is: $$r^2 - 2r = 0$$ Factor: $$r(r - 2) = 0$$ So, $$r = 0 \quad \text{or} \quad r = 2$$ The homogeneous solution is: $$x_h(t) = C_1 + C_2 e^{2t}$$ 4. **Find a particular solution:** Since the right side is $$e^t (t - 3)$$, try a particular solution of the form: $$x_p(t) = e^t (A t + B)$$ 5. **Compute derivatives:** $$x_p' = e^t (A t + B) + e^t A = e^t (A t + B + A)$$ $$x_p'' = e^t (A t + B + A) + e^t (A) = e^t (A t + B + 2A)$$ 6. **Substitute into the differential equation:** $$x_p'' - 2 x_p' = e^t (A t + B + 2A) - 2 e^t (A t + B + A) = e^t \big[(A t + B + 2A) - 2(A t + B + A)\big]$$ Simplify inside brackets: $$A t + B + 2A - 2 A t - 2 B - 2 A = -A t - B$$ So, $$x_p'' - 2 x_p' = e^t (-A t - B)$$ 7. **Set equal to the right side:** $$e^t (-A t - B) = e^t (t - 3)$$ Divide both sides by $$e^t$$: $$-A t - B = t - 3$$ Equate coefficients: For $$t$$ terms: $$-A = 1 \Rightarrow A = -1$$ For constants: $$-B = -3 \Rightarrow B = 3$$ 8. **Write the particular solution:** $$x_p(t) = e^t (-t + 3)$$ 9. **General solution:** $$x(t) = x_h(t) + x_p(t) = C_1 + C_2 e^{2t} + e^t (-t + 3)$$ 10. **Apply initial conditions:** At $$t=0$$: $$x(0) = C_1 + C_2 e^0 + e^0 (0 + 3) = C_1 + C_2 + 3 = 2$$ So, $$C_1 + C_2 = -1$$ 11. **Find $$x'(t)$$:** $$x'(t) = 0 + 2 C_2 e^{2t} + \frac{d}{dt} \big(e^t (-t + 3)\big)$$ Use product rule: $$\frac{d}{dt} \big(e^t (-t + 3)\big) = e^t (-t + 3) + e^t (-1) = e^t (-t + 3 - 1) = e^t (-t + 2)$$ So, $$x'(t) = 2 C_2 e^{2t} + e^t (-t + 2)$$ 12. **Apply initial condition $$x'(0) = 0$$:** $$x'(0) = 2 C_2 e^0 + e^0 (0 + 2) = 2 C_2 + 2 = 0$$ So, $$2 C_2 = -2 \Rightarrow C_2 = -1$$ 13. **Find $$C_1$$:** From step 10: $$C_1 + (-1) = -1 \Rightarrow C_1 = 0$$ 14. **Final solution:** $$\boxed{x(t) = - e^{2t} + e^t (-t + 3)}$$