Question

Four professors at an undisclosed university receive grants to study global warming. The temperature of the world, measured in units that norma the current temperature at 1, is predicted to evolve according to the equation y'=y2-2 y, y(0)-1. The professors give the following opinions: Zweistein: The temperature will become infinite in a finite time. Clearly this ill be the end of the world. Dreistein: The world will continue to warm, but there wille be end of the world. But eventually it will get too hot for everyon Vierstein: The world is warming, but it will reach a new equilibrium. If you like hot weather, you will be ok. Einstein: All this talk about global warming is nonsense. The world is actually cooling. Who is right?

Answer 1

$\frac{dy}{dt}=y(y-2)\\\implies \frac{dy}{y(y-2)}=dt\\ \implies dt=\{\frac{-1}{2y}+\frac{1}{2(y-2)}\}dy$

Integrating we get

$t+C=\{\frac{-1}{2}\ln y+\frac{1}{2}\ln {(y-2)}\}\\ \implies t+C=\frac{1}{2}\ln (\frac{y-2}{y})$

where $C$ is a constant of integration.

Thus we have

$\frac{y-2}{y}=e^{2(t+c)}\\ \implies y=\frac{2}{1-e^{t+c}}$

Here $t$ denotes the time passed and $y$ denotes the temperature.

As $t\to \infty$

$e^{2(t+c)}\to \infty \\ \implies y=\frac{2}{1-e^{t+c}}\to 0$

Thus the temperature decreases with increasing time and hence the 4th option is correct.