To solve the differential equation \( y'' + 5y' + 6y = -10te^{5t} \), we can use the method of undetermined coefficients. Here’s a step-by-step outline:

1. **Solve the Homogeneous Equation:**
   The associated homogeneous equation is:
   \[
   y'' + 5y' + 6y = 0
   \]
   The characteristic equation is:
   \[
   r^2 + 5r + 6 = 0
   \]
   Factoring gives:
   \[
   (r + 2)(r + 3) = 0
   \]
   Thus, the roots are \( r = -2 \) and \( r = -3 \).

   The general solution to the homogeneous equation is:
   \[
   y_h(t) = C_1 e^{-2t} + C_2 e^{-3t}
   \]

2. **Find a Particular Solution:**
   For the non-homogeneous part \( -10te^{5t} \), we guess a particular solution of the form:
   \[
   y_p(t) = t(Ae^{5t} + Be^{5t})
   \]
   However, since \( e^{5t} \) is not a solution of the homogeneous equation, we can use:
   \[
   y_p(t) = t(Ae^{5t})
   \]

   To find the coefficients \( A \), we need to compute \( y_p' \) and \( y_p'' \).

3. **Differentiate:**
   \[
   y_p' = A(e^{5t} + 5te^{5t}) = Ae^{5t}(1 + 5t)
   \]
   \[
   y_p'' = A(5e^{5t} + 5e^{5t} + 25te^{5t}) = A e^{5t}(10 + 25t)
   \]

4. **Substitute into the Original Equation:**
   Now substitute \( y_p \), \( y_p' \), and \( y_p'' \) into the left-hand side of the differential equation:
   \[
   A e^{5t}(10 + 25t) + 5Ae^{5t}(1 + 5t) + 6t(Ae^{5t}) = -10te^{5t}
   \]

   Combine like terms and solve for \( A \).

5. **Solve for Constants:**
   Equate coefficients from both sides of the equation to solve for \( A \).

6. **General Solution:**
   The general solution is:
   \[
   y(t) = y_h(t) + y_p(t) = C_1 e^{-2t} + C_2 e^{-3t} + t(Ae^{5t})
   \]

If you need a detailed calculation of the coefficients or have specific values for \( C_1 \) and \( C_2 \), please let me know!

### Do you want details on any specific part or have any questions? 

Here are some related questions:
1. What are the steps to solve a homogeneous differential equation?
2. How do you apply the method of undetermined coefficients?
3. What is the significance of the characteristic equation?
4. Can you provide another example of a non-homogeneous differential equation?
5. How do you check if your solution is correct? 

**Tip:** Always check the initial or boundary conditions if they are provided to determine specific constants in your solution.

Discover how to solve the differential equation y'' + 5y' + 6y = -10te^{5t} using the method of undetermined coefficients. This guide covers the steps to find both the homogeneous and particular solutions, essential for understanding advanced differential equations. Ideal for students in Grades 11-12.

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