https://file.aiquickdraw.com/mathbot/file/1725448604145zjz5zv6f.jpg

The image asks to solve the given equation using the Cauchy-Euler method. Let's proceed with solving the equation:

The Cauchy-Euler differential equation typically has the form:

\[
a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \ldots + a_1 x y' + a_0 y = 0.
\]

From the image, the equation is:

\[
x^2 y'' - 2x y' + 2y = 0.
\]

### Step 1: Form the Auxiliary Equation

Assuming a solution of the form \( y = x^m \), we differentiate to get:

\[
y' = m x^{m-1}, \quad y'' = m (m-1) x^{m-2}.
\]

Substituting these into the differential equation:

\[
x^2 [m (m-1) x^{m-2}] - 2x [m x^{m-1}] + 2 [x^m] = 0,
\]

simplifies to:

\[
m (m-1) x^m - 2m x^m + 2 x^m = 0.
\]

Combining like terms:

\[
(m^2 - m - 2m + 2) x^m = 0,
\]

\[
(m^2 - 3m + 2) x^m = 0.
\]

### Step 2: Solve the Auxiliary Equation

The auxiliary equation is:

\[
m^2 - 3m + 2 = 0.
\]

Factoring this equation:

\[
(m - 1)(m - 2) = 0.
\]

So, the solutions are:

\[
m = 1 \quad \text{and} \quad m = 2.
\]

### Step 3: General Solution

Since we have distinct roots, the general solution is:

\[
y(x) = C_1 x^1 + C_2 x^2,
\]

or simply:

\[
y(x) = C_1 x + C_2 x^2.
\]

Would you like further details or have any questions?

### Follow-Up Questions:

1. How does the Cauchy-Euler method compare to solving linear differential equations with constant coefficients?
2. What happens when the roots of the auxiliary equation are complex?
3. How do you modify the solution when the roots of the auxiliary equation are repeated?
4. Can you solve a Cauchy-Euler equation where \(x = 0\) is within the interval of interest?
5. How would the solution change if there were non-homogeneous terms in the equation?

### Tip:

For Cauchy-Euler equations, always check the form of the differential equation to ensure it matches \(x^n y^{(n)}\) terms, allowing direct use of the method.

Solve the equation x^2 y'' - 2xy' + 2y = 0 using the Cauchy-Euler method.

Step-by-step solution of the differential equation x^2 y'' - 2xy' + 2y = 0 using the Cauchy-Euler method. This problem explores key concepts in differential equations and is suitable for undergraduate mathematics students.

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