Homework for §1.5
Find the general solutions of the differential equations below. If an initial condition
is given, find the corresponding particular solution. Primes denote derivatives with
respect to x.
1. y 0 − 2y = 3e2x , y(0) = 0.
2. xy 0 + 5y = 7x2 , y(2) = 5.
3. (1 + x)y 0 + y = cos x, y(0) = 1.
.............................................................................
Solve the following differential equation below by regarding y as the independent
variable rather than x.
dy
=1
4. (x + yey )
dx
.............................................................................
5. A tank initially contains 60 gal of pure water. Brine containing 1 lb of salt per
gallon enters the tank at 2 gal/min, and the (perfectly mixed) solution leaves the
tank at 3 gal/min; thus the tank is empty in exactly 1 hour. (a) Find the amount
of salt in the tank after t minutes. (b) What is the maximum amount of salt ever in
the tank?
6. Consider a cascade of two tanks with V1 = 100 gal and V2 = 200 gal the volumes
of brine in the two tanks. Each tank also initially contains 50 lb of salt. Pure water
flows into tank 1 at a rate of 5 gal/sec, the mixture in tank 1 flows into tank 2 at
a rate of 5 gal/sec, and the mixture in tank 2 flows out at 5 gal/sec. (a) Find the
amount x(t) of salt in tank 1 at time t. (b) Suppose that y(t) is the amount of salt
in tank 2 at time t. Show first that
5x
5y
dy
=
−
,
dt
100 200
and then solve for y(t), using the function x(t) found in part (a). Finally, find the
maximum amount of salt ever in tank 2.