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Chapter 6: Problem 137

If \(y=x \sin x\), then prove that, \(x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+\left(x^{2}+2\right) y=0\)

Short Answer

Expert verified
After performing the differentiation and simplification processes, the equation \(x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+\left(x^{2}+2\right) y=0\) simplifies to \(0=0\). Thus, the given expression is proven.

Step by step solution

01

Differentiate the function \(y = x \sin x\)

Using the product rule stated as \((uv)'=u'v+uv'\), where \(u=x\) and \(v=\sin x\), the derivative becomes: \(\frac{dy}{dx} = \sin x + x \cos x\)
02

Differentiate \(\frac{dy}{dx}\) again

Applying product rule and sum rule for differentiation, this yields: \(\frac{d虏y}{dx虏} = \cos x + \cos x - x \sin x = 2 \cos x - x \sin x\)
03

Substitute into the differential equation and simplify

Substitute \(y, \frac{dy}{dx}, \frac{d虏y}{dx虏}\) into \(x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+\left(x^{2}+2\right) y=0\), it becomes: \(x^{2} (2 \cos x - x \sin x) - 2x (\sin x + x \cos x) + (x^{2}+2) x \sin x = 0\). After simplification, all terms cancel and the statement is reduced to \(0=0\)

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