1. **State the problem:** Determine the properties of the differential equation $$y'' + ty' + 4y = 0$$ from the given options. 2. **Recall definitions:** - A differential equation is **linear** if the dependent variable and its derivatives appear to the first power and are not multiplied together. - It is **homogeneous** if the equation equals zero (no standalone function of the independent variable). - The **order** of a differential equation is the highest derivative present. 3. **Analyze the given equation:** - The equation is $$y'' + ty' + 4y = 0$$. - The highest derivative is $$y''$$, which is the second derivative, so it is a **second order** differential equation. - The equation is equal to zero, so it is **homogeneous**. - The terms $$y''$$, $$ty'$$, and $$4y$$ are all linear in $$y$$ and its derivatives (no powers or products of $$y$$ or its derivatives). 4. **Conclusion:** - The equation is **linear**. - The equation is **homogeneous**. - The equation is **second order**. 5. **Answer choice:** - a. Linear - b. Homogeneous - d. Second order DE Since the question asks to select the correct property, the best single choice describing the equation is **a. Linear** (or b or d depending on the question format). Usually, the equation is all of these, but if only one choice is allowed, linearity is fundamental. Final answer: The differential equation $$y'' + ty' + 4y = 0$$ is **linear**, **homogeneous**, and **second order**.