JL Sem2_2013/2014

TMS2033 Differential Equations

SOLUTION TUTORIAL #2

1. Find the solution of the given initial value problem

a. 22 , (1) 0ty y te y

b. 2(2 / ) (cos ) / , ( ) 0, 0y t y t t y t

c. 3 24 , ( 1) 0, 0tt y t y e y t

2. Consider the initial value problem

0

33 2 , (0) .

2

ty y t e y y

Find the value of 0y that separates solutions that grow positively as t from

those that grow negatively. How does the solution that corresponds to this critical

value of 0y behave as t ?

JL Sem2_2013/2014

3. Solve the differential equation

a. 2 sin 0y y x

b. 2 2(cos )(cos 2 )y x y

c. 2 3/ (1 )y x y x

4. Solve the initial value problem

22(1 )(1 ), (0) 0y x y y

and determine where the solution attains its minimum value.

5. Suppose that a tank containing a certain liquid has an outlet near the bottom. Let h(t)

be the height of the liquid surface above the outlet at time t. Torricelli’s states that the

outflow velocity v at the outlet is equal to the velocity if a particle falling freely (with

no drag) from the height h.

a. Show that 2v gh , where g is the acceleration due to gravity.

b. By equating the rate of outflow to the rate of change of liquid in the tank,

show that h(t) satisfies the equation

JL Sem2_2013/2014

( ) 2 ,dh

A h a ghdt

where A(h) is the area of the cross section of the tank at height h and a is the area

of the outlet. The constant α is a contraction coefficient that accounts for the

observed fact that the cross section of the (smooth) outflow stream is smaller than

a. The value of α for water is about 0.6.

c. Consider a water tank in the form of a right circular cylinder that is 3m high

above the outlet. The radius of the tank is 1m and the radius of the circular

outlet is 0.1m. If the tank is initially full of water, determine how long it takes

to drain the tank down to the level of the outlet.

JL Sem2_2013/2014

6. A young person with no initial capital invests k dollars per year at an annual rate of

return r. Assume that investments are made continuously and that the return is

compounded continuously.

a. Determine the sum S(t) accumulated at any time t.

b. If r = 7.5%, determine k so that $1 million will be available for retirement in

40 years.

c. If k = $2000/year, determine the return rate r that must be obtained to have $1

million available in 40 years.

7. The population of mosquitoes in a certain area increases at a rate proportional to the

current population, and in the absence of other factors, the population doubles each

week. There are 200,000 mosquitoes in the area initially, and predators (birds, bats,

and so forth) eat 20,000 mosquitoes/day. Determine the population of mosquitoes in

the area at any time.

8. Without solving the problem, determine an interval in which the solution of the given

initial value problem is certain to exist.

a. ( 4) 0, (2) 1t t y y y

JL Sem2_2013/2014

b. (tan ) sin , ( ) 0y t y t y

c. 2 2(4 ) 2 3 , (1) 3t y ty t y

9. Solve the given initial value problem and determine how the interval in which the

solution exists depends on the initial value 0y .

a. 2

02 , (0)y ty y y

b. 3

00, (0)y y y y

10. The equation below occurs in the study of the stability of fluid flow.

3, 0 and 0y y y

Since this equation is a Bernoulli equation, you are to solve it by making a change

of the dependent variable that converts it into a linear equation.