Question

If \(y=(\tan^{-1} x)^2\), show that \((x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1)\frac{dy}{dx} = 2\).

Hint:

When dealing with proofs involving derivatives, it's often easier to rearrange the equation after the first differentiation to remove denominators. This allows you to use the simpler product rule instead of the more complex quotient rule for the second derivative.

Answer 1

Step 1: Understanding the Concept:
This problem involves second-order differentiation. The goal is to find the first and second derivatives of the given function and then show that they satisfy the given differential equation.
Step 2: Key Formula or Approach:
We will use the chain rule for the first derivative and the product rule for the second derivative. 1. Find \(\frac{dy}{dx}\).
2. Rearrange the equation to avoid fractions, which simplifies the next differentiation.
3. Differentiate again with respect to \(x\) using the product rule.
4. Rearrange the resulting equation to match the required form.
Step 3: Detailed Explanation:
We are given the function: \[ y = (\tan^{-1} x)^2 \] Differentiating with respect to \(x\) using the chain rule: \[ \frac{dy}{dx} = 2(\tan^{-1} x)^1 \cdot \frac{d}{dx}(\tan^{-1} x) \] \[ \frac{dy}{dx} = 2(\tan^{-1} x) \cdot \frac{1}{1+x^2} \] To avoid using the quotient rule for the next step, multiply both sides by \((1+x^2)\): \[ (1+x^2) \frac{dy}{dx} = 2 \tan^{-1} x \] Now, differentiate both sides again with respect to \(x\). We use the product rule on the left-hand side (LHS).
\[ \frac{d}{dx} \left[ (1+x^2) \frac{dy}{dx} \right] = \frac{d}{dx} (2 \tan^{-1} x) \] \[ (1+x^2) \frac{d}{dx}\left(\frac{dy}{dx}\right) + \frac{dy}{dx} \frac{d}{dx}(1+x^2) = 2 \cdot \frac{1}{1+x^2} \] \[ (1+x^2) \frac{d^2y}{dx^2} + \frac{dy}{dx} (2x) = \frac{2}{1+x^2} \] To eliminate the fraction on the right-hand side, multiply the entire equation by \((1+x^2)\): \[ (1+x^2) \left[ (1+x^2) \frac{d^2y}{dx^2} + 2x \frac{dy}{dx} \right] = (1+x^2) \left( \frac{2}{1+x^2} \right) \] \[ (x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1) \frac{dy}{dx} = 2 \] This is the required expression. Hence, it is shown. Step 4: Final Answer:
By differentiating the function \(y=(\tan^{-1} x)^2\) twice and performing algebraic manipulations, we have shown that \((x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1)\frac{dy}{dx} = 2\).