https://file.aiquickdraw.com/mathbot/file/1725843658353okblth9h.jpg

The image you provided contains some mathematical content that seems to involve a few key equations and a table.

### Extracted Content:

1. **Equation 1**: 
   \[
   f'(t) = - \frac{5}{t^{2}}f(t)
   \]
   This equation likely describes a first-order ordinary differential equation (ODE).

2. **Equation 2**: 
   \[
   f(t) = \frac{k}{t^{5}}
   \]
   This appears to be the general solution to the first ODE where \(k\) is a constant.

3. **Instructions**:
   - There is a question or problem statement asking the reader to compare this general solution with another form given specific conditions.
   - The table included suggests that there are some values (or solutions) being compared, likely involving constants \( k_1 \) and \( k_2 \) corresponding to different scenarios.

4. **Table Description**:
   - The table contains results or conditions for different values of \( C_{a} \) and \( C_{b} \) when \( k_1 = 0.1 \) and \( k_2 = 2.0 \).

### Steps to Understand and Solve the Problem:

1. **Interpret the Differential Equation**:
   - Identify what the differential equation represents and how its solution was derived.
   - If the given solution \( f(t) = \frac{k}{t^{5}} \) is indeed the general solution, substitute it back into the original ODE to confirm.

2. **Compare Solutions**:
   - If comparing two solutions (e.g., \( k_1 \) and \( k_2 \)), analyze how the constants affect the behavior of the solution.
   - Use the table to determine the impact of different \( C_a \) and \( C_b \) values on the constants.

3. **Verification**:
   - Verify the solutions by applying initial or boundary conditions if provided or implied by the table.
   - Understand the context of \( C_a \) and \( C_b \) in relation to the differential equation and its solution.

Would you like a detailed step-by-step analysis of the problem or further clarification on specific aspects? 

### Related Questions:

1. What is the general method to solve first-order differential equations like the one given?
2. How does changing the constant \( k \) affect the general solution \( f(t) \)?
3. What role do the parameters \( C_a \) and \( C_b \) play in the problem, based on the table provided?
4. How can the validity of the general solution \( f(t) = \frac{k}{t^{5}} \) be confirmed by substitution?
5. How would you apply specific initial conditions to find the exact value of \( k \)?

**Tip:** When dealing with differential equations, always check your solution by differentiating and substituting it back into the original equation to ensure it satisfies the equation.

The original problem involves solving a differential equation and comparing solutions for different values of constants based on a table.

Explore how to solve first-order differential equations with the equation f'(t) = -5/t^2 * f(t) and compare different constants using f(t) = k/t^5. This problem includes steps to verify the solution by substitution and an analysis of how constants affect the solution.

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