Sol tut01

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 0yyy
dt
dy


JL Sem2_2013/2014
5. Consider the differential equation
,
dy
ay b
dt
   where both a and b are positive numbers.
a. Solve the differential equation.
b. Sketch the solution for several different initial conditions.

JL Sem2_2013/2014
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 equationQ 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; ( ) , ( ) cosht
y y y t e y t t   
b. 2 2
; 3ty y t y t t   
c. 1 24 3 ; ( ) /3, ( ) /3iv t
y 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 2
2 1/2 /4
; ( / ) , 0x t
xx tu u u t e t
  
  

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