Chapter 4.P4, Problem 3P

Problems $1$ through $3$ are concerned with one degree-of-freedom systems.

The Brachistrochrone Problem. Find the curve $y\left(x\right)$ connecting $y\left(a\right)=A$ to $y\left(b\right)=B$ (assume $a<b$ and $B<A$), along which a particle under the influence of gravity $g$ will slide without friction in minimum time. This is the famous Brachistochrone problem discussed in Problem $31$ of section $2.2$. Bernoulli showed that the appropriate functional for the sliding time is

$T={\displaystyle \underset{0}{\overset{T}{\int }}dt}={\displaystyle \underset{0}{\overset{L}{\int }}\frac{dt}{ds}ds}={\displaystyle \underset{0}{\overset{L}{\int }}\frac{ds}{v}=\frac{1}{\sqrt{2g}}{\displaystyle \underset{0}{\overset{L}{\int }}\frac{ds}{\sqrt{y}}}}$

$=\frac{1}{\sqrt{2g}}{\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{\frac{1+{y^{\prime }}^2}{y}}dx}$ $\left(\text{i}\right)$

(a) Justify each of the equalities in Eq. $\left(\text{i}\right)$.

(b) From the functional on the right in Eq. $\left(\text{i}\right)$, find the Euler-Lagrange equation for the curve $y\left(x\right)$.

Consider the initial value problem

$y^{\prime }-\frac{3}{2}y=3t+2e^t,y\left(0\right)=y_0$.

Find the value of $y_0$ that separates solutions that grow positively as $t\to \infty$ from those that grow negatively. How does the solution that corresponds to this critical value of $y_0$ behave as $t\to \infty$?