Oxford University MAT 2018 Solved Papers

Question : 2 of 10

Marks: +1, -0

The function

y=e^{k x}

satisfies the equation

(\;{ {d}^2 y}/{ {d} x^2}+\;{ {d} y}/{ {d} x})(\;{ {d} y}/{ {d} x}-y)=y \;{ {d} y}/{ {d} x}

for

Solution:

Substitute in

y=e^{k x}

and use the fact that the derivative of

e^{k x}

k e^{k x}

. This gives

(k^2 e^{k x}+k e^{k x})(k e^{k x}-e^{k x})=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=√{2}

k=-√{2}

. The answer is (d).

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