Mathematics

Mathematics

Problem

Let y(x)=(x2−5x)sin⁡xy(x)=(x^{2}-5x)\sin x. Does yy satisfy the differential equation dydx=(2x−5)sin⁡x\dfrac{dy}{dx}=(2x-5)\sin x? a) Yes; the derivative is (2x−5)sin⁡x(2x-5)\sin x exactly. b) No; differentiating introduces an additional (x2−5x)cos⁡x(x^{2}-5x)\cos x term. c) Yes; using the identity sin⁡2x+cos⁡2x=1\sin^{2}x+\cos^{2}x=1 shows the extra term vanishes, leaving (2x−5)sin⁡x(2x-5)\sin x.},{ d) No; differentiating yields (2x−5)cos⁡x+(x2−5x)sin⁡x(2x-5)\cos x + (x^{2}-5x)\sin x, which does not match (2x−5)sin⁡x(2x-5)\sin x.

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