91Ó°ÊÓ

The temperature on the surface of a steel ball at time \(t\) is given by \(u(t)=70 e^{-k t}+\) 30 (in \({ }^{\circ} \mathrm{F}\) ) where \(k\) is a positive constant. Show that \(u\) satisfies the first order equation \(d u / d t=-k(u-30)\). What is the initial temperature \((t=0)\) on the surface of the ball? What happens to the temperature as \(t \rightarrow \infty\) ?

The displacement (measured from \(x=0\) ) of a mass attached to the end of a spring at time \(t\) is given by \(x(t)=\frac{1}{4} e^{-t}(\cos \sqrt{35} t+\) \(\frac{9}{\sqrt{35}} \sin \sqrt{35} t\) ). Show that \(x\) satisfies the ordinary differential equation \(x^{\prime \prime}+2 x^{\prime}+36 x=0\). What is the initial displacement of the mass? What is the initial velocity of the mass?

\(\frac{d y}{d x}=\frac{x-x^{2}}{(x+1)\left(x^{2}+1\right)}\)

\(x^{2} y^{\prime \prime}+3 x y^{\prime}+5 y=0, y(1)=0, y^{\prime}(1)=1\), \(y(x)=x^{-1}(A \cos (2 \ln x)+B \sin (2 \ln x))\)

Show that \(u(x, y)=\ln \sqrt{x^{2}+y^{2}}\) satisfies Laplace's equation, \(u_{x x}+u_{y y}=0\).