Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 17 / b / i

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- Solution i

Solution

In the homogeneous equation, substitute the exponential function

e^{k x}

. The characteristic equation is

k^{5} - k = k (k^{4} - 1) = k (k - 1) (k + 1) (k^{2} + 1) = 0.

Its distinct roots are

0, 1, - 1, i, - i

. Combining the complex-conjugate pair into real sine and cosine gives the complementary function

y_{c} = C_{0} + C_{1} e^{x} + C_{2} e^{- x} + C_{3} cos x + C_{4} sin x .

Five linearly independent terms supply the required five homogeneous degrees of freedom.

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