Question

If \(y = \cos^{-1} x\), then show that \((1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0\).

Hint:

When asked to prove differential equations involving inverse trigonometric functions, a very common and effective technique is to differentiate once, rearrange to remove square roots (by squaring), and then differentiate again implicitly. This avoids complex applications of the quotient or chain rule on fractional exponents.

Answer 1

Step 1: Understanding the Concept:
This problem requires finding the second derivative of an inverse trigonometric function and then showing that it satisfies a given second-order differential equation. A useful strategy is to manipulate the first derivative to remove square roots before differentiating a second time.
Step 2: Key Formula or Approach:
1. Find the first derivative, \(\frac{dy}{dx}\). 2. Rearrange the equation to eliminate the square root, typically by squaring both sides. 3. Differentiate the rearranged equation implicitly with respect to x. 4. Simplify the resulting equation to match the required form.
Step 3: Detailed Explanation:
Given the function: \[ y = \cos^{-1} x \] Differentiate with respect to x to find the first derivative: \[ \frac{dy}{dx} = -\frac{1}{\sqrt{1-x^2}} \quad \text{---(1)} \] To make the second differentiation easier, let's rearrange this equation to remove the square root. \[ \sqrt{1-x^2} \frac{dy}{dx} = -1 \] Now, square both sides of the equation: \[ (1-x^2) \left(\frac{dy}{dx}\right)^2 = (-1)^2 = 1 \] Differentiate this entire equation with respect to x, using the product rule on the left side: \[ \frac{d}{dx}\left[(1-x^2)\right] \left(\frac{dy}{dx}\right)^2 + (1-x^2) \frac{d}{dx}\left[\left(\frac{dy}{dx}\right)^2\right] = \frac{d}{dx}(1) \] \[ (-2x) \left(\frac{dy}{dx}\right)^2 + (1-x^2) \left[2 \left(\frac{dy}{dx}\right) \frac{d^2y}{dx^2}\right] = 0 \] We can see that \(2\frac{dy}{dx}\) is a common factor. Since \(\frac{dy}{dx}\) is not generally zero (from equation 1), we can divide the entire equation by \(2\frac{dy}{dx}\): \[ -x \frac{dy}{dx} + (1-x^2) \frac{d^2y}{dx^2} = 0 \] Rearranging the terms to match the required format: \[ (1-x^2) \frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0 \] Step 4: Final Answer:
We have successfully derived the given differential equation. Hence proved.