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

$f''(x)=x(-\sin x) +\cos x +\cos x$
 

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

$ \Rightarrow x(-\sin x)+\cos x+\cos x+x\sin x+t\cos x=0$

$\Rightarrow 2\cos x+t\cos x = 0$
$\Rightarrow \cos x(t+2)=0$
$\Rightarrow t+2=0, t=-2$

edited by
5 votes

Hence, t = -2

2 votes
t=-2
0
how?
1 vote
We have f(x) = x sin x

β‡’ f'(x) = x cos x + sin x

β‡’ fβ€²β€²(x) = x (βˆ’ sin x ) + cos x + cos x = (βˆ’x sin x ) + 2 cos x

Now, it is given that f(x) = x sin x satisfies the equation fβ€²β€²(x) + f(x) + t cos x = 0

β‡’ (βˆ’x 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 = βˆ’2
Answer:

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