GATE CS 2014 Set 1 Q.56 (CS Q.46): Calculus

The function $f(x) = x \sin x$ satisfies the following equation: $f''(x) + f(x) + t \cos x = 0$ . The value of $t$ is ______.

Your answer

Official answer and solution

Solution

Compute the second derivative of $f(x) = x \sin x$ using the product rule and substitute into the given differential equation.

Given $f(x) = x \sin x$ .

First derivative by the product rule:

f'(x) = \frac{d}{dx}[x \sin x] = 1 \cdot \sin x + x \cdot \cos x = \sin x + x \cos x

Second derivative:

f''(x) = \frac{d}{dx}[\sin x + x \cos x] = \cos x + (1 \cdot \cos x - x \sin x) = 2 \cos x - x \sin x

Now substitute $f(x)$ and $f''(x)$ into the given equation:

f''(x) + f(x) + t \cos x = 0

(2 \cos x - x \sin x) + (x \sin x) + t \cos x = 0

2 \cos x + t \cos x = 0

(2 + t) \cos x = 0

Since this equation must hold for all $x$ :

2 + t = 0 \implies t = -2

Thus, the value of $t$ is $-2$ .

Final answer

-2

Common trap

Miscalculating the derivative of $x \cos x$ as $-x \sin x$ , dropping the $+\cos x$ product term.

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