The function y=e^{k x} satisfies the equation \,≤ft( d² y/d x² + d y/d x ) ≤ft( d y/d x - y ) = y \,d y/d x for

Correct Answer is : exactly three distinct values of kkk,

exactly two distinct values of k,

exactly three distinct values of k,

infinitely many distinct values of k.

Correct Answer is : exactly three distinct values of kkk,

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Oxford University MAT 2018 Solved Papers

Question : 2 of 10

Marks: +1, -0

The function

y=e^{k x}

\,\left( \frac{d^2 y}{d x^2} + \frac{d y}{d x} \right) \left( \frac{d y}{d x} - y \right) = y \,\frac{d y}{d x}

Solution:

Substitute in

y=e^{k x}

and use the fact that the derivative of

e^{k x}

k e^{k x}

. This gives

\left( k^2 e^{k x} + k e^{k x} \right) \left( k e^{k x} - e^{k x} \right) = k e^{2 k x}

Now

e^{k x}

is never zero, so we may divide each side by

e^{2 k x}

, giving an cubic equation for

k ; k(k+

(k-1)=k

. This has three solutions;

k=0

k=\sqrt{2}

k=-\sqrt{2}

. The answer is (d).

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