We are given the differential equation:

\[
(D^4 - 2D^3 + D^2)y = 0
\]

Here, \( D \) represents the differential operator, where \( D^n = \frac{d^n}{dx^n} \), meaning the \( n \)-th derivative with respect to \( x \). Let's rewrite this equation more explicitly:

\[
\left( \frac{d^4}{dx^4} - 2 \frac{d^3}{dx^3} + \frac{d^2}{dx^2} \right) y = 0
\]

This is a homogeneous linear differential equation with constant coefficients. The general approach is to solve the corresponding characteristic equation:

\[
r^4 - 2r^3 + r^2 = 0
\]

We can factor this equation:

\[
r^2(r^2 - 2r + 1) = 0
\]

Factor the quadratic part:

\[
r^2(r - 1)^2 = 0
\]

The roots of this characteristic equation are \( r = 0 \) (with multiplicity 2) and \( r = 1 \) (with multiplicity 2).

Now, the general solution to the differential equation is given by:

\[
y(x) = (C_1 + C_2x) + (C_3 + C_4x)e^x
\]

where \( C_1, C_2, C_3, \) and \( C_4 \) are arbitrary constants.

This solution reflects the fact that we have repeated roots, so we include the terms \( x \) and \( e^x \) multiplied by constants to account for the multiplicity.

Would you like further details or have any questions? 

Here are five related questions to expand on this topic:
1. How would the solution change if the characteristic equation had complex roots?
2. Can you explain the method of undetermined coefficients for non-homogeneous equations?
3. What is the significance of the multiplicity of roots in differential equations?
4. How does the general solution structure change for higher-order differential equations?
5. How does variation of parameters differ from solving homogeneous equations?

**Tip:** When solving higher-order differential equations, always start by finding the characteristic equation, as it gives you the framework for the solution.

Explore how to solve the differential equation (D^4 - 2D^3 + D^2)y = 0 using the characteristic equation method. This problem covers key concepts in differential equations, factorization techniques, and the general solution structure for repeated roots. Suitable for undergraduate students studying calculus and differential equations.

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