The function $f(x) = x \sin x$ satisfies the following equation: $f^{\prime\prime}(x) + f(x) + t \cos x = 0$. The value of $t$ is _____.

Given:

Compute the first derivative $f^{\prime}(x)$ using the product rule:

Compute the second derivative $f^{\prime\prime}(x)$:

Substitute $f(x)$ and $f^{\prime\prime}(x)$ into the given differential equation:

For this equation to hold for all $x$, we must have:

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