https://file.aiquickdraw.com/mathbot/file/17304336046142zs6lt1m.jpg

To solve the differential equation given in the image:

\[
y^{(5)} - 6y^{(4)} + 9y''' - 6y'' + 8y' = 0
\]

we'll proceed as follows:

1. **Form the Characteristic Equation:** For a linear differential equation of the form \( y^{(5)} - 6y^{(4)} + 9y''' - 6y'' + 8y' = 0 \), the corresponding characteristic equation is:
   \[
   r^5 - 6r^4 + 9r^3 - 6r^2 + 8r = 0
   \]

2. **Factor Out \( r \):**
   We can factor out \( r \) from the characteristic equation:
   \[
   r(r^4 - 6r^3 + 9r^2 - 6r + 8) = 0
   \]
   This gives one root, \( r = 0 \).

3. **Solve the Remaining Polynomial \( r^4 - 6r^3 + 9r^2 - 6r + 8 = 0 \):**
   We can try factoring or using the Rational Root Theorem to find other roots. In this case, we observe that it factors as:
   \[
   (r - 2)(r - 2)(r - 2)(r + 2) = 0
   \]
   or
   \[
   (r - 2)^3 (r + 2) = 0
   \]

   This gives the roots:
   - \( r = 0 \)
   - \( r = 2 \) (with multiplicity 3)
   - \( r = -2 \)

4. **Write the General Solution:** 
   Based on these roots, we can write the general solution for \( y(t) \) as follows:
   \[
   y(t) = c_1 + c_2 e^{2t} + c_3 t e^{2t} + c_4 t^2 e^{2t} + c_5 e^{-2t}
   \]
   where \( c_1, c_2, c_3, c_4, \) and \( c_5 \) are arbitrary constants.

### Final Answer:
\[
y(t) = c_1 + c_2 e^{2t} + c_3 t e^{2t} + c_4 t^2 e^{2t} + c_5 e^{-2t}
\]

Would you like a detailed breakdown of each step or have any questions?

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### Related Questions

1. How would you find the characteristic equation for a second-order linear differential equation?
2. What is the significance of the multiplicity of roots in the characteristic equation?
3. How does the presence of a zero root affect the solution of a differential equation?
4. How would you solve a differential equation if complex roots were present?
5. What methods can be used to factor higher-degree polynomials?

### Tip:
Remember, the multiplicity of a root in the characteristic equation affects the solution's form by introducing polynomial terms in \( t \) (e.g., \( t \), \( t^2 \)) for each repeated root.

Find the general solution of the differential equation: y^(5) - 6y^(4) + 9y''' - 6y'' + 8y' = 0.

Explore the general solution of the differential equation y^(5) - 6y^(4) + 9y''' - 6y'' + 8y'. This solution covers advanced concepts in differential equations, including characteristic equations, root multiplicity, and constructing solutions based on root types. Suitable for college-level students.

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