1. **State the problem:** Solve the differential equation $$(2D^2 - 5D - 3)y = 0$$ where $D$ represents the differential operator $\frac{d}{dx}$. 2. **Rewrite the equation:** The operator form means $$2\frac{d^2y}{dx^2} - 5\frac{dy}{dx} - 3y = 0$$. 3. **Characteristic equation:** Replace $D$ by $r$ to get the characteristic polynomial: $$2r^2 - 5r - 3 = 0$$. 4. **Solve the quadratic:** Use the quadratic formula: $$r = \frac{5 \pm \sqrt{(-5)^2 - 4 \times 2 \times (-3)}}{2 \times 2} = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm \sqrt{49}}{4}$$ 5. **Calculate roots:** $$r = \frac{5 \pm 7}{4}$$ So, - $$r_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3$$ - $$r_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$$ 6. **General solution:** Since roots are real and distinct, the solution is: $$y = C_1 e^{3x} + C_2 e^{-\frac{x}{2}}$$ where $C_1$ and $C_2$ are arbitrary constants determined by initial conditions.