NCEA Calculus 2024 TWO(c - Solution & Explanation

Question

Show that

y = \sin(x^2) - \cos(x)

is a solution to the equation

\dfrac{d^2y}{dx^2} + 4x^2\,y = 2\cos(x^2) + (1 - 4x^2)\cos x

.

Official Answer

\dfrac{dy}{dx}=2x\cos\left(x^2\right)+\sin x

\dfrac{d^2y}{dx^2}=2\cos\left(x^2\right)-2x\cdot2x\sin\left(x^2\right)+\cos x

\dfrac{d^2y}{dx^2}=2\cos\left(x^2\right)-4x^2\sin\left(x^2\right)+\cos x

Then LHS

=\dfrac{d^2y}{dx^2}+4x^2y

\qquad=2\cos\left(x^2\right)-4x^2\sin\left(x^2\right)+\cos x+4x^2\left(\sin\left(x^2\right)-\cos x\right)

\qquad=2\cos\left(x^2\right)+\cos x-4x^2\cos x

\qquad=2\cos\left(x^2\right)+\left(1-4x^2\right)\cos x

\qquad=\text{RHS}

As required.

Grading Criteria

Correct justification that it is a solution, with evidence of both $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ .

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Video Explanation

2024 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 22:02)

2024 TWO(c)