Solve

(

x

2

D

2

− xD

+

1

)

y

=

4 cos

(log

(

x

)

)

+

x

2

We are dealing with a second-order linear differential equation of the form:

\[
\left( x^2 D^2 - x D + 1 \right) y = 4 \cos(\log(x)) + x^2
\]

where \( D = \frac{d}{dx} \) represents the derivative with respect to \( x \).

### Step 1: Rewrite the equation
First, express the operator \( D \) in terms of derivatives:
\[
D y = \frac{dy}{dx}, \quad D^2 y = \frac{d^2 y}{dx^2}
\]
Thus, the given equation becomes:

\[
x^2 \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + y = 4 \cos(\log(x)) + x^2
\]

This is now a non-homogeneous second-order differential equation.

### Step 2: Solve the homogeneous equation
The associated homogeneous equation is:

\[
x^2 \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + y = 0
\]

This is a **Cauchy-Euler** equation, and the general solution to such equations is of the form:

\[
y_h = x^r
\]

Substitute \( y_h = x^r \) into the homogeneous equation:

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

Simplifying:

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

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

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

Thus, \( r = 1 \) is a repeated root. Since the root is repeated, the general solution to the homogeneous equation is:

\[
y_h = c_1 x + c_2 x \log(x)
\]

### Step 3: Solve the non-homogeneous equation
Now, we need to find a particular solution \( y_p \) to the non-homogeneous equation:

\[
x^2 \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + y = 4 \cos(\log(x)) + x^2
\]

Let's assume the particular solution consists of two parts, \( y_p = y_1 + y_2 \), where:
- \( y_1 \) corresponds to the \( 4 \cos(\log(x)) \) term.
- \( y_2 \) corresponds to the \( x^2 \) term.

#### 1. For \( y_1 \):

We try a solution of the form:

\[
y_1 = A \cos(\log(x)) + B \sin(\log(x))
\]

Substitute this into the left-hand side of the differential equation, and solve for \( A \) and \( B \). After differentiating and simplifying, you will find that:

\[
y_1 = -2 \cos(\log(x))
\]

#### 2. For \( y_2 \):

Since the right-hand side has an \( x^2 \) term, try:

\[
y_2 = C x^2
\]

Substitute this into the equation and solve for \( C \). You will find:

\[
y_2 = \frac{1}{2} x^2
\]

### Step 4: General solution

The general solution to the original equation is:

\[
y = y_h + y_p
\]

\[
y = c_1 x + c_2 x \log(x) - 2 \cos(\log(x)) + \frac{1}{2} x^2
\]

This is the complete solution.

Would you like further details on any specific step, or do you have any questions?

### Related Questions:
1. How do you solve Cauchy-Euler differential equations in general?
2. Why do repeated roots result in solutions involving logarithms?
3. How would you solve the equation if the non-homogeneous term were different?
4. What other methods can be used to solve non-homogeneous differential equations?
5. How does the method of undetermined coefficients work?

### Tip:
For non-homogeneous equations, identifying the form of the particular solution based on the right-hand side is crucial for applying the method of undetermined coefficients effectively.

Solve (x^2D^2 - xD + 1)y = 4 cos(log(x)) + x^2

Learn how to solve a second-order non-homogeneous Cauchy-Euler differential equation of the form (x^2D^2 - xD + 1)y = 4 cos(log(x)) + x^2. This step-by-step solution covers the method of undetermined coefficients, handling both trigonometric and polynomial terms. Suitable for undergraduate students.

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