https://file.aiquickdraw.com/mathbot/file/17350782511756fkofb8p.jpg

The differential equation provided in the image is:

\[
x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 3y = \frac{\log x}{x^2}.
\]

This is a second-order linear non-homogeneous differential equation. The general approach to solving this involves two steps:

1. Solve the corresponding **homogeneous equation**:
   \[
   x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 3y = 0.
   \]

2. Find a **particular solution** for the non-homogeneous equation:
   \[
   x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 3y = \frac{\log x}{x^2}.
   \]

### Step 1: Solve the Homogeneous Equation

We solve the homogeneous part using the substitution \(y = x^m\), where \(m\) is a constant to be determined. Substituting \(y = x^m\), we compute:
\[
\frac{dy}{dx} = mx^{m-1}, \quad \frac{d^2y}{dx^2} = m(m-1)x^{m-2}.
\]

Substituting these into the homogeneous equation gives:
\[
x^2 \left[m(m-1)x^{m-2}\right] + 5x \left[mx^{m-1}\right] + 3x^m = 0.
\]

Simplifying:
\[
m(m-1)x^m + 5mx^m + 3x^m = 0.
\]

Factor \(x^m\):
\[
x^m \left[m^2 + 4m + 3\right] = 0.
\]

The characteristic equation is:
\[
m^2 + 4m + 3 = 0.
\]

Factoring:
\[
(m+3)(m+1) = 0.
\]

Thus, the roots are \(m = -3\) and \(m = -1\). Therefore, the general solution to the homogeneous equation is:
\[
y_h = C_1 x^{-3} + C_2 x^{-1},
\]
where \(C_1\) and \(C_2\) are arbitrary constants.

---

### Step 2: Find the Particular Solution

The non-homogeneous term is \(\frac{\log x}{x^2}\). We use the method of undetermined coefficients to guess the form of the particular solution. Since the right-hand side involves \(\log x\), we propose:
\[
y_p = A \log x + B.
\]

Compute the derivatives of \(y_p\):
\[
\frac{dy_p}{dx} = \frac{A}{x}, \quad \frac{d^2y_p}{dx^2} = -\frac{A}{x^2}.
\]

Substitute \(y_p\), \(\frac{dy_p}{dx}\), and \(\frac{d^2y_p}{dx^2}\) into the original equation:
\[
x^2 \left(-\frac{A}{x^2}\right) + 5x \left(\frac{A}{x}\right) + 3(A \log x + B) = \frac{\log x}{x^2}.
\]

Simplify each term:
\[
-A + 5A + 3A \log x + 3B = \frac{\log x}{x^2}.
\]

Combine like terms:
\[
(3A) \log x + (4A + 3B) = \frac{\log x}{x^2}.
\]

For the equation to hold true, the coefficients of \(\log x\) and the constant term must separately match the right-hand side:
1. \(3A = \frac{1}{x^2} \implies A = \frac{1}{3x^2}.\)
2. \(4A + 3B = 0 \implies B = -\frac{4A}{3}.\)

Substitute \(A = \frac{1}{3x^2}\) into \(B\):
\[
B = -\frac{4}{3} \cdot \frac{1}{3x^2} = -\frac{4}{9x^2}.
\]

Thus, the particular solution is:
\[
y_p = \frac{\log x}{3x^2} - \frac{4}{9x^2}.
\]

---

### General Solution

The general solution is the sum of the homogeneous solution and the particular solution:
\[
y = y_h + y_p = C_1 x^{-3} + C_2 x^{-1} + \frac{\log x}{3x^2} - \frac{4}{9x^2}.
\]

Let me know if you need further clarification or additional details!

---

### Related Questions

1. How do you verify a solution to a differential equation?
2. What are the common methods for solving second-order linear equations?
3. Can the method of undetermined coefficients handle all non-homogeneous terms?
4. How would you approach this problem using the variation of parameters?
5. What are the applications of second-order differential equations in physics?

---

### Tip  
When solving second-order differential equations, always confirm the form of your proposed solution matches the nature of the non-homogeneous term!

Solve the second-order linear differential equation provided in the image.

Explore a step-by-step solution to the second-order linear differential equation x^2y'' + 5xy' + 3y = log(x)/x^2. Learn how to find the general solution by solving the homogeneous equation and determining the particular solution using the method of undetermined coefficients.

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