Exercise1 (16 pts)
A large tank is partially filled with 100 gallons of fluid in which 10 pounds of
salt is dissolved.Brine containing 0.5 pound of salt per gallon is pumped into
the tank at a rate 6gal/min. The well mixed solution is then pumped out at
a slower rate of 4gal/min.
1. Find the number of pounds of salt in the tank after 30 minutes.
2. Suppose that the tank is of size 500 gallons, when will the tank
overflow?
3. How many pounds of salt will be in the tank at the instant it
overflows?
Exercise 2 (20pts)
Classify each D.E (linear, separable, homogeneous, Bernouilli) then solve.
𝑥𝑦+2𝑦−𝑥−2
1. 𝑦 ′ = 𝑥𝑦−3𝑦+𝑥−3
2. (𝑒 𝑥 + 𝑒 −𝑥 )𝑦 ′ = 𝑦 2 ,
3. 𝑥 2 𝑦′ + 𝑥(𝑥 + 2)𝑦 = 𝑒 𝑥 ,
4. 𝑦 ′ = 𝑦(𝑥𝑦 3 − 1),
𝑦
5.
𝑦
(𝑥 + 𝑦𝑒 𝑥 ) 𝑑𝑥 = 𝑥𝑒 𝑥 𝑑𝑦.
Exercise 3 (8pts)
Find the value(s) of k so that the given differential equation is exact
(2𝑥 − 𝑦 sin(𝑥𝑦) + 𝑘𝑦 4 )𝑑𝑥 − (20 𝑥𝑦 3 + 𝑥 sin(𝑥𝑦))𝑑𝑦 = 0.
Exercise 4 (28 pts)
Let the autonomous D.E
𝑦 ′ = (𝑦 − 3)2 (𝑦 2 + 𝑦 − 2)
(E)
1. Identify the equilibrium solutions and classify them (source/unstable
sink/stable; node/semi-stable).
2. Skech a phase line of (E).
3. The direction field of (E) is given by the figure below
use the direction field above from the differential equation (E) to
sketch the graph of the solution for the given initial conditions.
a) Y(0)=3
b) Y(0)=2
c) Y(0)=-1
Exercise 5 (12pts)
Estimate the following solution using Euler’s method with n=5 steps over
the interval [0,1]
𝑦 ′ = 3𝑡 − 𝑦
{
𝑦(0) = 1
Compare your solution with the exact solution.
Exercise 6 (16 pts)
The D.E
𝑦 ′ = 𝑃(𝑥) + 𝑄(𝑥)𝑦 + 𝑅(𝑥)𝑦 2
Is known as Ricatti’s equation. A Ricatti equation can be solved by a
succession of two substitutions provided we know a particular solution 𝑦1
of the equation. First use the substitution 𝑦 = 𝑦1 + 𝑢, and then discuss the
equation satisfied by u.
a) Find one parameter family of solutions for the D.E
4 1
𝑦′ = − 2 − 𝑦 + 𝑦2
𝑥
𝑥
2
Where 𝑦1 = is a known solution of the equation.
𝑥