The given differential equation is:

\[
y' = x \left(x^2 + b^2\right)^{-\frac{1}{2}}
\]

To find the general solution \( y(x) \), we need to integrate both sides with respect to \( x \).

### Step 1: Simplify the equation
The differential equation can be rewritten as:

\[
y' = \frac{x}{\sqrt{x^2 + b^2}}
\]

### Step 2: Integrate both sides
Integrate \( y' \) with respect to \( x \):

\[
y = \int \frac{x}{\sqrt{x^2 + b^2}} \, dx
\]

To solve this integral, we can use the substitution:

Let \( u = x^2 + b^2 \), so \( du = 2x \, dx \). Therefore, the integral becomes:

\[
y = \frac{1}{2} \int \frac{du}{\sqrt{u}}
\]

This simplifies to:

\[
y = \frac{1}{2} \cdot 2\sqrt{u} + C = \sqrt{x^2 + b^2} + C
\]

### Step 3: Write the final solution
The general solution to the differential equation is:

\[
y(x) = \sqrt{x^2 + b^2} + C
\]

where \( C \) is the constant of integration.

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Would you like further details or have any questions? 

Here are five related questions:
1. What would happen if \( b = 0 \)?
2. How does the solution change if the differential equation has a different power?
3. Can you derive the solution if \( b \) is a function of \( x \)?
4. How would you solve the differential equation using a different method, like separation of variables?
5. What are the real-world applications of such a differential equation?

**Tip:** Remember that substitution is a powerful tool for solving integrals, especially when dealing with square roots or trigonometric functions.

Learn how to solve the differential equation y' = x(x^2 + b^2)^(-1/2) step-by-step. This problem involves differential equations, integration by substitution, and applications of trigonometric functions. Suitable for undergraduate level study.

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