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The given equation is:

\[
x^2 y'' + 19xy' + 81y = 0
\]

with the initial conditions \( y(1) = 0 \) and \( y'(1) = 1 \).

This is a second-order linear differential equation with variable coefficients. To solve it, we recognize that it has the form of a **Cauchy-Euler** (or **Euler-Cauchy**) equation. The general approach to solve such equations is by assuming a solution of the form:

\[
y(x) = x^r
\]

### Step 1: Substitute \( y(x) = x^r \) into the differential equation.

- \( y = x^r \)
- \( y' = r x^{r-1} \)
- \( y'' = r(r-1) x^{r-2} \)

Substitute these into the equation:

\[
x^2 \left( r(r-1) x^{r-2} \right) + 19x \left( r x^{r-1} \right) + 81x^r = 0
\]

Simplify each term:

\[
r(r-1) x^r + 19r x^r + 81 x^r = 0
\]

Factor out \( x^r \):

\[
x^r \left( r(r-1) + 19r + 81 \right) = 0
\]

Since \( x^r \neq 0 \), we have the characteristic equation:

\[
r(r-1) + 19r + 81 = 0
\]

### Step 2: Solve the characteristic equation.

\[
r^2 - r + 19r + 81 = 0
\]
\[
r^2 + 18r + 81 = 0
\]

Solve this quadratic equation using the quadratic formula:

\[
r = \frac{-18 \pm \sqrt{18^2 - 4(1)(81)}}{2(1)}
\]
\[
r = \frac{-18 \pm \sqrt{324 - 324}}{2}
\]
\[
r = \frac{-18 \pm 0}{2}
\]
\[
r = -9
\]

Since the solution has a repeated root \( r = -9 \), the general solution for the Cauchy-Euler equation in this case is:

\[
y(x) = (C_1 + C_2 \ln x) x^{-9}
\]

### Step 3: Apply the initial conditions.

1. \( y(1) = 0 \):

\[
y(1) = (C_1 + C_2 \ln 1) \cdot 1^{-9} = C_1 = 0
\]

So, \( C_1 = 0 \).

2. \( y'(1) = 1 \):

First, find \( y'(x) \):

\[
y(x) = C_2 x^{-9} \ln x
\]
\[
y'(x) = C_2 \left( -9x^{-10} \ln x + \frac{x^{-10}}{x} \right)
= C_2 x^{-10} (-9 \ln x + 1)
\]

Substitute \( x = 1 \):

\[
y'(1) = C_2 \cdot 1^{-10} \cdot (-9 \ln 1 + 1) = C_2 \cdot 1 = C_2
\]

Given \( y'(1) = 1 \), we have \( C_2 = 1 \).

### Final Solution:

Thus, the solution is:

\[
y(x) = x^{-9} \ln x
\]

Would you like further details or have any questions?

Here are 5 related questions for you to explore:
1. How does the solution change if the characteristic equation has distinct roots?
2. Can you solve this equation using a power series method?
3. What is the physical interpretation of Cauchy-Euler equations in applied mathematics?
4. How would the solution differ if the initial condition \( y'(1) = 0 \) were changed?
5. What are other types of second-order linear differential equations with variable coefficients?

**Tip:** When solving Cauchy-Euler equations, always look for a characteristic equation to reduce the complexity.

Find y as a function of x if y(1) = 0, y'(1) = 1, and x^2 y'' + 19xy' + 81y = 0.

Learn how to solve the second-order differential equation x^2 y'' + 19xy' + 81y = 0 with initial conditions y(1) = 0 and y'(1) = 1. This problem involves solving a Cauchy-Euler equation using characteristic equations and the quadratic formula. Suitable for university-level students studying calculus and differential equations.

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