The given equation is:

$2xy + 3 = 5x^2$

We are asked to find $x \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx}$.

Step 1: Implicit Differentiation

First, let's differentiate the equation $2xy + 3 = 5x^2$ implicitly with respect to $x$.

First derivative:

$\frac{d}{dx} (2xy + 3) = \frac{d}{dx} (5x^2)$ Applying the product rule on $2xy$, we get: $2 \left( y + x \frac{dy}{dx} \right) = 10x$ Simplifying, we get: $2y + 2x \frac{dy}{dx} = 10x$ $y + x \frac{dy}{dx} = 5x$

Second derivative:

Now, let's differentiate the equation $y + x \frac{dy}{dx} = 5x$ again with respect to $x$.

$\frac{d}{dx} \left( y + x \frac{dy}{dx} \right) = \frac{d}{dx} (5x)$ Applying the product rule again on $x \frac{dy}{dx}$, we get: $\frac{dy}{dx} + \left( \frac{dy}{dx} + x \frac{d^2 y}{dx^2} \right) = 5$ Simplifying: $2 \frac{dy}{dx} + x \frac{d^2 y}{dx^2} = 5$

Step 2: Rearrange the equation

The expression $x \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx}$ is already simplified, and it equals:

$x \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} = 5$

Thus, the final result is:

$x \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} = 5$

Would you like a detailed explanation of any step?

Here are 5 related questions you might find helpful:

1. What is implicit differentiation, and how is it used?
2. Can we solve for $y$ explicitly in terms of $x$ from the original equation?
3. What are some applications of second-order derivatives in real-world problems?
4. How do product and chain rules help in differentiation of composite functions?
5. Can you derive the second derivative from different types of implicit functions?

Tip: Always double-check when applying the product rule in differentiation, especially when variables are multiplied.