91Ó°ÊÓ

Verify that \(y=\sin x, y=\cos x, y=e^{i x}\), and \(y=e^{-i x}\) are all solutions of \(y^{\prime \prime}=-y\).

Solve the following differential equations. \(\left(2 D^{2}+D-1\right) y=0\)

By separation of variables, solve the differential equation \(d y / d x=\sqrt{1-y^{2}}\) to obtain solution containing one arbitrary constant. Although this solution may be referred to as the " general solution," show that \(y=1\) is a solution of the differential equation not obtainablc from the "general solution" by any choice of the arbitrary constant. The solution \(y=1\) is called a singular selution; \(y=-1\) is another singular solution. Sketch a number of graphs of the "general solution" for different values of the arbitrary constant and observe that \(y=1\) is tangent to all of them. This is characteristic of a singular solution -its graph is tangent at each point to one of the graphs of the "general solution." Note that the given differential equation is not linear; for linear equations, all solutions are contained in the general solution, but nonlinear equations may have singular solutions which cannot be obtained from the "general solution" by specializing the arbitrary constant (or constants). Thus a nonlinear first-order equation in \(x\) and \(y\) may have two (or more) solutions passing through a given point in the \((x, y)\) plane, whereas a linear first-order equation always has just one such solution. Show that any continuous curve made up of pieces of \(y=1, y=-1\), and the sinc curves of the "general solution," gives a solution of the above differential equation. Sketch such a solution curve on your graphs.

Find the orthogonal trajectories of each of the following families of curves. In each case sketch several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from \(d y / d x\) for the original curves; this constant takes different values for different curves of the original family and you want an expression for \(d y / d x\) which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. \(y=k x^{2}\)

A simple pendulum consists of a point mass \(m\) suspended by a weightless cord of length \(l .\) Find the equation of motion of the pendulum, that is, the differential equation for \(\theta\) as a function of \(t .\) Show that (for small \(\theta\) ) this is approximately a simple harmonic motion equation, and find \(\theta\) if \(\theta=\theta_{0}, d \theta / d t=0\) when \(t=0\).

A block of wood is floating in water; it is depressed slightly and then released to oscillate up and down. Assume that the top and bottom of the block are parallel planes which remain horizontal during the oscillations and that the sides of the block are vertical. Show that the period of the motion (neglecting friction) is \(2 \pi \sqrt{h / g}\), where \(h\) is the vertical height of the part of the block under water when it is floating at rest. Hint: Recall that the buoyant force is equal to the weight of displaced water.