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Chapter 1: Problem 5
\(\left(\frac{d y}{d x}\right)^{2}+y=0\)
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(a) Show that \(\left(x^{2}+y^{2}\right)^{2}=5 x y\) is an implicit solution of $$ \begin{aligned} &{\left[4 x\left(x^{2}+y^{2}\right)-5 y\right] d x} \\ &\quad+\left[4 y\left(x^{2}+y^{2}\right)-5 x\right] d y=0 \end{aligned} $$ (b) Graph \(\left(x^{2}+y^{2}\right)^{2}=5 x y\) on the rectangle \([-2,2] \times[-2,2]\). (c) Approximate all points on the graph of \(\left(x^{2}+y^{2}\right)^{2}=5 x y\) with \(x\)-coordinate 1 . (d) Approximate all points on the graph of \(\left(x^{2}+y^{2}\right)^{2}=5 x y\) with \(y\)-coordinate \(-0.319\).
\(y^{\prime \prime \prime}-4 y^{\prime}=0, y(0)=1, y^{\prime}(0)=-1, y^{\prime \prime}(0)=0\), \(y=A+B e^{2 x}+C e^{-2 x}\)
For a particular wire of length 1 foot, the temperature at time \(t\) hours at a position of \(x\) feet from the end \((x=0)\) of the wire is estimated by \(u(x, t)=e^{-\pi^{2} k t} \sin \pi x-e^{-4 \pi^{2} k t} \times\) \(\sin 2 \pi x\). Show that \(u\) satisfies the equation \(u_{t}=k u_{x x}\). What is the initial temperature \((t=0)\) at \(x=1\) ? What happens to the temperature at each point in the wire as \(t \rightarrow \infty\) ?
\(\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}, c>0\) constant
\(d y / d x=x^{-2} \cos \left(x^{-1}\right), y(2 / \pi)=1\)
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