JL Sem2_2013/2014

TMS2033 Differential Equations

SOLUTION TUTORIAL#1

1. Draw the direction field for the given differential equations. Based on the direction

field, determine the behavior of y as t . Describe the dependency of its solution

on the initial value of y at t = 0.

a. y’ = 3 – 2y

b. y’ = 3 + 2y

c. y’ = 1 + 2y

JL Sem2_2013/2014

d. y’ = -y(5 - y)

e. y’ = y(y – 2)

2

2. Write down a differential equation of the form dy/dt = ay + b whose solutions have

the required behavior as t .

a. All solutions approach y = 2/3

JL Sem2_2013/2014

b. All other solutions diverge from y = 2

c. All other solutions diverge from y = 1/3

3. A certain drug is being administered intravenously to a hospital patient. Fluid

containing 5mg/cm3 of the drug enters the patient’s bloodstream at a rate of

100cm3/hr. The drug is absorbed by body tissues or otherwise leaves the bloodstream

at a rate proportional to the amount present, with a rate constant of 0.4(hr)-1

.

a. Assuming that the drug is always uniformly distributed throughout the

bloodstream, write a differential equation for the amount of the drug that is

present in the bloodstream at any time.

b. How much of the drug is present in the bloodstream after a long time?

4. Solve the following initial value problem and plot the solutions for several values of

y0. Describe briefly how the solutions resemble, and differ from, each other.

.)0(,5 0yyydt

dy

JL Sem2_2013/2014

5. Consider the differential equation

,dy

ay bdt

where both a and b are positive numbers.

a. Solve the differential equation.

b. Sketch the solution for several different initial conditions.

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c. Describe how the solutions change under each of the following conditions:

i. a increases

ii. b increases

iii. Both a and b increases, but the ratio b/a remains the same.

6. Consider a population p of field mice that grows at a rate proportional to the current

population, so that dp/dt = rp.

a. Find the rate constant r if the population doubles in 30 days

b. Find r if the population doubles in N days

7. The half-life of a radioactive material is the time required for an amount of this

material to decay to one-half its original value. Show that for any radioactive material

that decays according to the equation Q rQ , the half-life τ and the decay rate r

satisfy the equation rτ = ln2.

8. A pool containing 1,000,000 gal of water is initially free of a certain undesirable

chemical. Water containing 0.01g/gal of the chemical flows into the pond at a rate of

300gal/hr, and water also flows out of the pond at the same rate. Assume that the

chemical is uniformly distributed throughout the pond.

a. Let Q(t) be the amount of the chemical in the pond at time t. Write down an

initial value problem for Q(t).

JL Sem2_2013/2014

b. Solve the problem in part (a) for Q(t). How much chemical is in the pond after

1 year?

c. At the end of 1 year the source of the chemical in the pond is removed;

thereafter pure water flows into the pond, and the mixture flows out at the

same rate as before. Write down the initial value problem that describes this

new situation.

d. Solve the initial value problem in part (c). How much chemical remains in the

pond after 1 additional year (2 years from the beginning of the problem)?

e. How long does it take for Q(t) to be reduced to 10g?

JL Sem2_2013/2014

f. Plot Q(t) versus t for 3 years

9. Verify that each given function is a solution of the differential equation

a. 1 20; ( ) , ( ) coshty y y t e y t t

b. 2 2; 3ty y t y t t

c. 1 24 3 ; ( ) / 3, ( ) / 3iv ty y y t y t t y t e t

10. Verify that each given function is a solution of the given partial differential equation

JL Sem2_2013/2014

a. 2

1 2; ( , ) sin sin , ( , ) sin( ), a real constantxx tta u u u x t x at u x t x at

b. 2 22 1/ 2 / 4; ( / ) , 0x t

xx tu u u t e t