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Chapter 6: Problem 120

If \(y=2 \sin x+3 \cos x\), prove that, \(\frac{d^{2} y}{d x^{2}}+y=0\)

Short Answer

Expert verified
Yes, the function \(y = 2 \sin x + 3 \cos x\) does indeed satisfy the equation \(\frac{d^{2} y}{d x^{2}}+y=0\). This is confirmed by following the rules of differentiation and subsequent addition of the derived results.

Step by step solution

01

- Differentiation

Start by differentiating the function \(y=2 \sin x+3 \cos x\) with respect to \(x\). Using the rules of differentiation, the derivative of \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\), we find: \( \frac{dy}{dx} = 2 \cos x - 3 \sin x\).
02

- Second Differentiation

Next, differentiate this result again with respect to \(x\). Once again applying the rules of differentiation we obtain: \(\frac{d^{2} y}{dx^{2}} = -2 \sin x - 3 \cos x\).
03

- Final Proof

To prove the given relation, sum \(y\) and \(\frac{d^{2} y}{dx^{2}}\), and check whether it equals zero: \[y + \frac{d^{2} y}{dx^{2}} = (2 \sin x+3 \cos x) + (-2 \sin x - 3 \cos x) = 0\]. This equality verifies the result, hence the proof.

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