Solve differential equations with hyperbolics

5

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$2 \cosh ^ { 2 } x = \cosh 2 x + 1$$ \includegraphics[max width=\textwidth, alt={}, center]{d421652f-576d-4843-abbf-54404e225fec-08_67_1550_374_347}
2. Find the solution of the differential equation $$\frac { d y } { d x } + 2 y \tanh x = 1$$ for which \(y = 1\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).

6 In this question you must show detailed reasoning.} The power output, \(p\) watts, of a machine at time \(t\) hours after it is switched on can be modelled by the equation \(\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )\) for \(t \geqslant 0\). Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to \(\mathbf { 2 }\) decimal places.

7 The diagram shows a curve which starts from the point \(A\) with coordinates ( 0,2 ). The curve is such that, at every point \(P\) on the curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } s$$ where \(s\) is the length of the \(\operatorname { arc } A P\).
\includegraphics[max width=\textwidth, alt={}, center]{587aac5c-fbc2-41d2-b1b3-16f3f7851d9d-4_399_764_1324_605}

1. 1. Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \sqrt { 4 + s ^ { 2 } }$$ (3 marks)
   2. Hence show that $$s = 2 \sinh \frac { x } { 2 }$$
   3. Hence find the cartesian equation of the curve.
2. Show that $$y ^ { 2 } = 4 + s ^ { 2 }$$