1. **Stating the problem:** We are given the differential equation $$t^{2} \cdot \frac{d^{2}x}{dt^{2}} + t \cdot x \cdot \frac{dx}{dt} = 0$$ and need to characterize it. 2. **Identify the order:** The highest derivative present is $$\frac{d^{2}x}{dt^{2}}$$, so this is a second-order differential equation. 3. **Check linearity:** A differential equation is linear if the dependent variable and its derivatives appear to the power 1 and are not multiplied together. 4. **Examine terms:** Here, the term $$t \cdot x \cdot \frac{dx}{dt}$$ involves the product of $$x$$ and $$\frac{dx}{dt}$$, which makes the equation nonlinear. 5. **Check homogeneity:** The equation equals zero, so it is homogeneous. 6. **Summary:** - Order: second-order - Nonlinear (due to product of $$x$$ and $$\frac{dx}{dt}$$) - Homogeneous **Final characterization:** The differential equation is a second-order, nonlinear, homogeneous differential equation.