A student attempts to verify that ( ) is a solution of the differential equation dy…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A student attempts to verify that y=x2ln⁡xy=x^{2}\ln x (x>0x>0) is a solution of the differential equation dydx=2xln⁡x+x2.\frac{dy}{dx}=2x\ln x+\frac{x}{2}. The student calculates dydx=2xln⁡x\dfrac{dy}{dx}=2x\ln x and concludes that the function fails to satisfy the equation. (a) Explain why the derivative calculated by the student is incomplete. (b) Find the correct expression for dydx\dfrac{dy}{dx} for y=x2ln⁡xy=x^{2}\ln x. (c) Using your result from part (b), determine whether y=x2ln⁡xy=x^{2}\ln x satisfies the differential equation for x>0x>0. Justify your answer.

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