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2013 Paper 3 Q7
D: 1700.0 B: 1500.0
differential equation second order energy method conservation law differentiation inequality hyperbolic functions Lyapunov function

  1. Let \(y(x)\) be a solution of the differential equation \( \dfrac {\d^2 y}{\d x^2}+y^3=0\) with \(y = 1\) and \(\dfrac{\d y}{\d x} =0\) at \(x=0\), and let \[ {\rm E} (x)= \left ( \frac {\d y}{\d x}\right)^{\!\!2} + \tfrac 12 y^4\,. \] Show by differentiation that \({\rm E}\) is constant and deduce that \( \vert y(x) \vert \le 1\) for all \(x\).
  2. Let \(v(x)\) be a solution of the differential equation \( \dfrac{\d^2 v}{\d x^2} + x \dfrac {\d v}{\d x} +\sinh v =0\) with \(v = \ln 3\) and \(\dfrac{\d v}{\d x} =0\) at \(x=0\), and let \[ {\rm E} (x)= \left ( \frac {\d v}{\d x}\right)^{\!\!2} + 2 \cosh v\,. \] Show that \(\dfrac{\d{\rm E}}{\d x}\le 0\) for \(x\ge0\) and deduce that \(\cosh v(x) \le \frac53\) for \(x\ge0\).
  3. Let \(w(x)\) be a solution of the differential equation \[ \frac{\d^2 w}{\d x^2} + (5\cosh x - 4 \sinh x -3) \frac{\d w}{\d x} + (w\cosh w + 2 \sinh w) =0 \] with \(\dfrac{\d w }{\d x}=\dfrac 1 { \sqrt 2 }\) and \(w=0\) at \(x=0\). Show that \(\cosh w(x) \le \frac54\) for \(x\ge0\).

Solution:

  1. \(\,\) \begin{align*} && E(x) &= \left ( \frac{\d y}{\d x} \right)^2 + \frac12 y^4 \\ \Rightarrow && E'(x) &= 2 \frac{\d y}{\d x} \frac{\d^2 y}{\d x^2} + 2y^3 \frac{\d y}{\d x} \\ &&&= 2\frac{\d y}{\d x} \left ( \frac{\d^2 y}{\d x^2} + y^3 \right) \\ &&&= 0 \end{align*} Therefore \(E\) is constant. \(E(0) = \frac12\) and \begin{align*} &&\frac12 &= \left ( \frac{\d y}{\d x} \right)^2 + \frac12 y^4 \\ &&&\geq \frac12 y^4 \\ \Rightarrow && |y| &\leq 1 \end{align*}
  2. \(\,\) \begin{align*} && E(x) &= \left ( \frac{\d v}{\d x} \right)^2 + 2 \cosh v \\ \Rightarrow && E'(x) &= 2 \frac{\d v}{\d x}\frac{\d^2 v}{\d^2 x} + 2 \sinh v \frac{\d v}{\d x} \\ &&&= 2 \frac{\d v}{\d x} \left ( \frac{\d^2 v}{\d^2 x} + \sinh v\right) \\ &&&= 2 \frac{\d v}{\d x} \left ( -x \frac{\d v}{\d x}\right) \\ &&&= -2x \left ( \frac{\d v}{\d x} \right)^2 \leq 0 \tag{\(x \geq 0\)} \\ \\ && E(0) &= 0^2 + 2 \cosh \ln 3 \\ &&&= 3 + \frac13 = \frac{10}{3} \\ \Rightarrow && \frac{10}{3} &\geq E(x) \\ &&&= \left ( \frac{\d v}{\d x} \right)^2 + 2 \cosh v \\ &&&\geq 2 \cosh v \\ \Rightarrow && \cosh v &\leq \frac53 \end{align*}
  3. \(\,\) \begin{align*} && E(x) &= \left ( \frac{\d w}{\d x} \right)^2 + 2(w \sinh w + \cosh w) \\ && E'(x) &= 2 \frac{\d w}{\d x}\frac{\d^2 w}{\d^2 x} + 2(w \cosh w + 2 \sinh w) \frac{\d w}{\d x} \\ &&&= 2 \frac{\d w}{\d x} \left ( \frac{\d^2 w}{\d^2 x}+(w \cosh w + 2 \sinh w)\right) \\ &&&= -2 \left ( \frac{\d w}{\d x} \right)^2 \left (\underbrace{5\cosh x - 4 \sinh x -3}_{\geq0} \right) \\ &&&\leq 0 \\ && E(0) &= \frac12 + 2 = \frac52 \\ \Rightarrow && \frac52 &\geq E(x) \\ &&&=\underbrace{ \left ( \frac{\d w}{\d x} \right)^2}_{\geq0} + 2(\underbrace{w \sinh w}_{\geq 0} + \cosh w) \\ &&&\geq2\cosh w \\ \Rightarrow && \cosh w &\leq \frac54 \end{align*}

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