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To solve this problem, we need to prove that:

\[
4(1 + x^2) \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} - 9y = 0
\]

given that:

\[
y = \left( x + \sqrt{1 + x^2} \right)^{3/2}
\]

### Step 1: Differentiate \( y \) with respect to \( x \)

Let
\[
u = x + \sqrt{1 + x^2}
\]
so that
\[
y = u^{3/2}
\]

First, find \( \frac{dy}{dx} \) using the chain rule:
\[
\frac{dy}{dx} = \frac{3}{2} u^{1/2} \frac{du}{dx}
\]

Now, calculate \( \frac{du}{dx} \):
\[
\frac{du}{dx} = 1 + \frac{x}{\sqrt{1 + x^2}} = \frac{\sqrt{1 + x^2} + x}{\sqrt{1 + x^2}} = \frac{1 + x^2 + x}{\sqrt{1 + x^2}}
\]

This gives us

If \( y = \left( x + \sqrt{1 + x^2} \right)^{3/2} \), show that \( 4(1 + x^2) \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} - 9y = 0 \).

Explore the step-by-step solution to prove that \( 4(1 + x^2) \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} - 9y = 0 \) given \( y = \left( x + \sqrt{1 + x^2} \right)^{3/2} \). This problem covers key calculus concepts, including the chain rule, implicit differentiation, and product rule. Ideal for advanced high school calculus students.

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