4 Find the particular solution of the differential equation
$$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$
given that, when \(x = 0 , y = 2\) and \(\frac { d y } { d x } = - 8\).
Starting from the definitions of cosh and sinh in terms of exponentials, prove that
$$\sinh 2 x = 2 \sinh x \cosh x$$
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Using the substitution \(\mathrm { u } = \sinh \mathrm { x }\), find \(\int \sinh ^ { 2 } 2 x \cosh x \mathrm {~d} x\).
Find the particular solution of the differential equation
$$\frac { d y } { d x } + y \tanh x = \sinh ^ { 2 } 2 x$$
given that \(y = 4\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
State the sum of the series \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 }\), for \(z \neq 1\).
By letting \(z = \cos \theta + i \sin \theta\), where \(\cos \theta \neq 1\), show that
$$1 + \cos \theta + \cos 2 \theta + \ldots + \cos ( n - 1 ) \theta = \frac { 1 } { 2 } \left( 1 - \cos n \theta + \frac { \sin n \theta \sin \theta } { 1 - \cos \theta } \right)$$
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The diagram shows the curve with equation \(\mathrm { y } = \cos \mathrm { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
By considering the sum of the areas of these rectangles, show that
$$\int _ { 0 } ^ { 1 } \cos x d x < \frac { 1 } { 2 n } \left( 1 - \cos 1 + \frac { \sin 1 \sin \frac { 1 } { n } } { 1 - \cos \frac { 1 } { n } } \right)$$
Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \cos x d x\).
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