To find the general solution of the non-homogeneous ordinary differential equation (ODE)

\[
(1+x^2)y''(x) + xy'(x) - y(x) = \sqrt{1+x^2},
\]

we can approach it by solving the corresponding homogeneous equation first, then finding a particular solution to the non-homogeneous equation.

### Step 1: Solve the Homogeneous Equation

The homogeneous equation associated with the given ODE is:

\[
(1+x^2)y''(x) + xy'(x) - y(x) = 0.
\]

This is a second-order linear differential equation with variable coefficients. One way to solve this is to use a trial solution \( y_h(x) = x^n \). Substituting \( y_h(x) = x^n \) into the homogeneous equation gives:

\[
(1+x^2)\cdot n(n-1)x^{n-2} + x\cdot n x^{n-1} - x^n = 0.
\]

Simplifying, we get:

\[
n(n-1)x^n + n x^n - x^n = 0,
\]

\[
[n(n-1) + n - 1]x^n = 0,
\]

\[
[n^2 - 1]x^n = 0.
\]

This simplifies to:

\[
n^2 - 1 = 0,
\]

\[
n = \pm 1.
\]

Thus, the general solution to the homogeneous equation is:

\[
y_h(x) = C_1 x + C_2 \frac{1}{x}.
\]

### Step 2: Find a Particular Solution

To solve the non-homogeneous equation, we now look for a particular solution \( y_p(x) \) to the full equation. Given the form of the right-hand side \( \sqrt{1+x^2} \), a reasonable guess for \( y_p(x) \) is:

\[
y_p(x) = A \sqrt{1+x^2}.
\]

Substituting \( y_p(x) = A \sqrt{1+x^2} \) into the original equation:

\[
(1+x^2) \left[\frac{d^2}{dx^2} \left(A \sqrt{1+x^2}\right)\right] + x \frac{d}{dx} \left(A \sqrt{1+x^2}\right) - A \sqrt{1+x^2} = \sqrt{1+x^2}.
\]

First, we find the first and second derivatives of \( y_p(x) \):

\[
y_p'(x) = \frac{A x}{\sqrt{1+x^2}},
\]

\[
y_p''(x) = \frac{A}{(1+x^2)^{3/2}}.
\]

Substituting into the equation:

\[
(1+x^2)\cdot \frac{A}{(1+x^2)^{3/2}} + x \cdot \frac{A x}{\sqrt{1+x^2}} - A \sqrt{1+x^2} = \sqrt{1+x^2}.
\]

This simplifies to:

\[
\frac{A}{\sqrt{1+x^2}} + \frac{A x^2}{\sqrt{1+x^2}} - A \sqrt{1+x^2} = \sqrt{1+x^2}.
\]

\[
\frac{A(1 + x^2)}{\sqrt{1+x^2}} - A \sqrt{1+x^2} = \sqrt{1+x^2}.
\]

This simplifies further:

\[
0 = (1-A)\sqrt{1+x^2}.
\]

Thus, \( A = 1 \).

Therefore, the particular solution is:

\[
y_p(x) = \sqrt{1+x^2}.
\]

### Step 3: Write the General Solution

The general solution of the non-homogeneous equation is the sum of the homogeneous solution and the particular solution:

\[
y(x) = y_h(x) + y_p(x) = C_1 x + C_2 \frac{1}{x} + \sqrt{1+x^2}.
\]

This is the general solution to the given non-homogeneous ODE.

---

Would you like any further details or explanations on any part of the solution?

Here are 8 related questions you might consider:
1. How do you solve second-order linear ODEs with constant coefficients?
2. What is the method of undetermined coefficients?
3. How do you find the particular solution when the right-hand side is a polynomial?
4. Can you use variation of parameters to solve this ODE?
5. What is the significance of the Wronskian in solving ODEs?
6. How can you apply initial conditions to find a specific solution?
7. What are the conditions for a differential equation to have a unique solution?
8. How do you verify the solution to an ODE?

**Tip:** For non-homogeneous ODEs, always start by solving the associated homogeneous equation before finding a particular solution.

Learn how to solve the non-homogeneous ordinary differential equation (ODE) (1+x^2)*y''(x) + x*y'(x) - y = sqrt(1+x^2) step-by-step. This problem involves second-order differential equations, homogeneous and non-homogeneous equations, and the trial solution method. Suitable for advanced undergraduate students and beyond.

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