Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 10] Higher-Order Differential Equations 345
is
2908*. The air
proportional
resistance to a body falling with a parachute
to the square of the rate of fall. Find the limiting
velocity of descent.
2909*. The bottom of a tank
is covered with a mixture of salt
with a capacity of 300 litres
and some insoluble substance.
Assuming that the rate at which the salt dissolves is proportional
to Ihe difference between the concentration at the given time
and the concentration of a saturated solution (1 kg of salt per 3
litres of water) and that the given quantity of pure water dissolves
1/3 kg of salt in 1 minute, find the quantity of salt in solution
at the expiration of one hour.
2910*. The electromotive force e in a circuit with current i,
resistance /? and self-induction L is made up of the voltage drop
Rl and the electromotive force of self-induction L^. Determine
the current / at time / if e^Esmat (E and o> are constants)
and i = when = 0.
Sec. 10. Higher-Order Differentia) Equations
then
1. The case of direct integration. If
n
i Miles
2. Cases of reduction of order. I) If a differential equation does not
contain y explicitly, for instance,
then, assuming y' p, we get an equation
F(x, p t p')-0.
ot an order one unit lower;
Example I. Find the particular solution of the equation
that satisfies the conditions
^ = 0, f/' = when x = 0.
Solution. Putting #'=p, we have / = p', whence
Solving the latter equation as a linear equation in the function p,
we get