Problem 4: Solving the Euler-Cauchy Equation
Given:

x^3 y''' - 2x^2 y'' - 2xy' + 8y = 0

Given the Euler-Cauchy equation:

\[
x^3 y''' - 2x^2 y'' - 2xy' + 8y = 0
\]

We solve this equation using the method specific to Euler-Cauchy equations, assuming a solution of the form \( y(x) = x^r \).

### Step 1: Find the derivatives
Start by computing the derivatives of \( y(x) = x^r \):

\[
y' = r x^{r-1}, \quad y'' = r(r-1) x^{r-2}, \quad y''' = r(r-1)(r-2) x^{r-3}
\]

### Step 2: Substitute into the equation
Substitute these derivatives into the original equation:

\[
x^3 \cdot r(r-1)(r-2) x^{r-3} - 2x^2 \cdot r(r-1) x^{r-2} - 2x \cdot r x^{r-1} + 8x^r = 0
\]

Simplify each term:

\[
r(r-1)(r-2) x^r - 2r(r-1) x^r - 2r x^r + 8x^r = 0
\]

Factor out \( x^r \):

\[
x^r \left[ r(r-1)(r-2) - 2r(r-1) - 2r + 8 \right] = 0
\]

### Step 3: Simplify the characteristic equation
The expression inside the brackets forms the characteristic equation:

\[
r(r-1)(r-2) - 2r(r-1) - 2r + 8 = 0
\]

Expand and simplify:

\[
r^3 - 3r^2 + 2r - 2r^2 + 2r - 2r + 8 = 0
\]

Combine like terms:

\[
r^3 - 5r^2 + 8 = 0
\]

### Step 4: Solve the characteristic equation
Now, we solve the cubic characteristic equation:

\[
r^3 - 5r^2 + 8 = 0
\]

This cubic equation can be solved either by factoring or using numerical methods. We can try factoring by testing potential roots (like 1, 2, etc.) or apply the cubic formula.

Let's test \( r = 2 \):

\[
(2)^3 - 5(2)^2 + 8 = 8 - 20 + 8 = -4
\]

\( r = 2 \) is not a root. Let's test \( r = 1 \):

\[
(1)^3 - 5(1)^2 + 8 = 1 - 5 + 8 = 4
\]

\( r = 1 \) is also not a root. Since we don't have an easy root by testing, we resort to numerical methods or approximate factorization to find the roots.

### Step 5: General Solution
Assuming you solve the cubic equation and find the roots \( r_1 \), \( r_2 \), and \( r_3 \), the general solution to the differential equation is:

\[
y(x) = C_1 x^{r_1} + C_2 x^{r_2} + C_3 x^{r_3}
\]

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

If you want, I can numerically approximate the roots or use a specific method to determine the exact roots. Let me know how you'd like to proceed!

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**Related Questions:**
1. How do you factorize or solve cubic equations analytically?
2. What are the implications of complex roots in differential equations?
3. How does the solution change if the roots are repeated?
4. Can this equation be solved using a series solution approach?
5. How do you verify the solution to a differential equation?

**Tip:** Solving cubic equations can be tricky; when roots aren't obvious, numerical methods or a calculator can help find approximate solutions quickly.

Learn how to solve the Euler-Cauchy equation x^3 y''' - 2x^2 y'' - 2xy' + 8y = 0 step-by-step with a detailed solution method. This problem involves advanced differential equations, specifically the Euler-Cauchy equation, and requires understanding derivatives and characteristic equations. Suitable for advanced undergraduate students and enthusiasts of mathematical analysis.

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