If $y=\mathrm{A} \sin x+\mathrm{B} \cos x$, then prove that $\frac{d^{2} y}{d x^{2}}+y=0$.

Step 1: Differentiate the given equation with respect to $x$

We start by finding the first derivative of the given function $y$ with respect to $x$. We apply the basic differentiation rules: the derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$. The constants A and B remain as coefficients.

Step 2: Differentiate the first derivative again with respect to $x$

Next, we find the second derivative by differentiating the first derivative, $\frac{d y}{d x}$, with respect to $x$. Again, we apply the differentiation rules: the derivative of $\cos x$ is $-\sin x$, and the derivative of $\sin x$ is $\cos x$.

Step 3: Substitute the second derivative into the expression

Now we substitute the expression for $\frac{d^{2} y}{d x^{2}}$ that we just found, and the original expression for $y$, into the equation we need to prove: $\frac{d^{2} y}{d x^{2}}+y=0$.

Step 4: Simplify the expression

By combining like terms, we can see that the terms $-\mathrm{A} \sin x$ and $\mathrm{A} \sin x$ cancel each other out, and similarly, $-\mathrm{B} \cos x$ and $\mathrm{B} \cos x$ cancel each other out. This results in a sum of zero, thus proving the given statement.

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