To determine the intervals on which a solution of the differential equation

$(t-4)^{2}y^{(3)} + \frac{7}{8} e^{4t} y^{(2)} - \sqrt{5-t} y^{(1)} + 2y = 0$

is guaranteed to exist, we need to analyze the coefficients of the equation.

Analysis of Coefficients:

1. Coefficient of $y^{(3)}$: $(t-4)^{2}$

   • This term is zero when $t = 4$. Thus, there could be issues at this point.

2. Coefficient of $y^{(2)}$: $\frac{7}{8} e^{4t}$

   • This term is always positive for all $t$.

3. Coefficient of $y^{(1)}$: $-\sqrt{5-t}$

   • This term is defined only when $t \leq 5$ and becomes zero at $t = 5$.

4. Coefficient of $y$: $2$

   • This term is always positive.

Intervals Evaluation:

• (0, 5]: Possible, as all coefficients are defined and continuous.
• [5, ∞): Not valid since $-\sqrt{5-t}$ is undefined for $t > 5$.
• (4, ∞): Not valid since $(t-4)^{2}$ is zero at $t = 4$.
• (4, 5]: Possible, as all coefficients are defined and continuous in this interval.
• (0, 4): Possible, as all coefficients are defined and continuous in this interval.
• (0, ∞): Not valid because $-\sqrt{5-t}$ is not defined for $t > 5$.

Conclusion:

The intervals where a solution is guaranteed to exist are:

• (0, 5]
• (4, 5]
• (0, 4)

Summary:

The correct selections are:

1. (0, 5]
2. (4, 5]
3. (0, 4)

Would you like to go over any specific points or have further questions? Here are some related inquiries:

1. What is the significance of the existence of solutions in differential equations?
2. Can you explain the role of continuity in determining the existence of solutions?
3. How do boundary conditions affect the solutions of differential equations?
4. What methods can be used to solve this differential equation?
5. How does the order of the differential equation impact the existence of solutions?

Tip: Always check the coefficients of the differential equation for points of discontinuity or where they become zero, as this often indicates where solutions may not exist.