**1.1)** A physical phenomenon is described by the quantities pressure *P*, length *l*, mass *m*, time *t* och density . If there is a physical law

relating these quantities, show that there is an equivalent physical law on the form

** **

**1.2)** Let

be a given continuous function. If *M=*max*|u(t)|* then *u* can be scaled with *M* to the dimensionless variable U=u/M. As time scale we can use *N=M/*max*|u'(t)|*, the ratio between the maximal value of the function and its maximal slope. Find *M* and *N* for the the following functions:

*a)
*

*b)
*

*c)*

**1.3)** Consider the function

where is a small number. Use exercise **1.2** to find a time scale. Is this time scale suitable for all the interval [0,1]? (Draw a graph of *u(t)* when ). Explain why two different time scales might be needed for a process described by *u(t)*.

**1.4)** A pendulum that is l long and has mass m swings around its point of fixing. We denote the angle between the vertical line and the pendulum with . This means that is 0 when the pendulum is directed straight down.

Use Newton’s second law to derive its equation of motion (note that the acceleration of the mass is *d ^{2}s/dt^{2}* where is the length of the circular arc that the mass travels along, that the force is

If the mass is released from a small angle at time *t=0*, formulate a dimensionless initial value problem that describes the motion.

**1.5)** Present and discuss Malthus’ logistic model for population dynamics. Introduce dimensionless variables and make a suitable scaling. Solve the scaled equation and find an attractor to the original equation. What does this attractor mean in practice?