1. **State the problem:** Solve the differential equation $\left(x^2D^2 + xD\right)y = 0$, where $D = \frac{d}{dx}$. 2. **Rewrite the equation:** The operator $D$ represents differentiation with respect to $x$. So the equation is: $$x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} = 0$$ 3. **Divide through by $x$ (assuming $x \neq 0$):** $$x \frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$$ 4. **Let $p = \frac{dy}{dx}$, then $\frac{d^2y}{dx^2} = \frac{dp}{dx}$:** $$x \frac{dp}{dx} + p = 0$$ 5. **This is a first-order linear ODE in $p$: ** $$x \frac{dp}{dx} = -p$$ 6. **Rewrite as:** $$\frac{dp}{dx} = -\frac{p}{x}$$ 7. **Separate variables:** $$\frac{dp}{p} = -\frac{dx}{x}$$ 8. **Integrate both sides:** $$\int \frac{1}{p} dp = -\int \frac{1}{x} dx$$ $$\ln|p| = -\ln|x| + C_1$$ 9. **Exponentiate:** $$|p| = e^{C_1} \frac{1}{|x|}$$ Let $C = e^{C_1}$, so $$p = \frac{C}{x}$$ 10. **Recall $p = \frac{dy}{dx}$, so:** $$\frac{dy}{dx} = \frac{C}{x}$$ 11. **Integrate again:** $$y = C \int \frac{1}{x} dx = C \ln|x| + C_2$$ 12. **General solution:** $$y = C_1 \ln|x| + C_2$$ **Answer:** The general solution to the differential equation is $$y = C_1 \ln|x| + C_2$$