The function \(f(x) =x \sin x\) satisfies the following equation: \(f''(x) +…
2014
The function \(f(x) =x \sin x\) satisfies the following equation:
\(f''(x) + f(x) +t \cos x = 0\)
The value of \(t\) is______.
Attempted by 31 students.
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Correct answer: -2
Solution: Compute derivatives and substitute into the given equation.
Compute the first derivative: f'(x) = sin x + x cos x.
Compute the second derivative: f''(x) = 2 cos x - x sin x.
Substitute into the equation f''(x) + f(x) + t cos x = 0: (2 cos x - x sin x) + (x sin x) + t cos x = (2 + t) cos x = 0.
Since this must hold for all x, the coefficient of cos x must be zero, so 2 + t = 0, giving t = -2.
Answer: t = -2