how?

14 votes

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

27 votes

Best answer

1 vote

We have f(x) = x sin x

β f'(x) = x cos x + sin x

β fβ²β²(x) = x (β sin x ) + cos x + cos x = (βx sin x ) + 2 cos x

Now, it is given that f(x) = x sin x satisfies the equation fβ²β²(x) + f(x) + t cos x = 0

β (βx sin x ) + 2 cos x + x sin x + t cos x = 0

β 2 cos x + t cos x = 0

β cos x ( t + 2 ) = 0

β t + 2 = 0

β t = β2

β f'(x) = x cos x + sin x

β fβ²β²(x) = x (β sin x ) + cos x + cos x = (βx sin x ) + 2 cos x

Now, it is given that f(x) = x sin x satisfies the equation fβ²β²(x) + f(x) + t cos x = 0

β (βx sin x ) + 2 cos x + x sin x + t cos x = 0

β 2 cos x + t cos x = 0

β cos x ( t + 2 ) = 0

β t + 2 = 0

β t = β2