Math 3C Final Practice Questions
1. Consider the following differential equation:
dy
 t  ty
dt
a)
b)
c)
d)
What are the constant solutions to this differential equation, if any?
Classify any constant solutions as stable or unstable.
Is this differential equation linear? Is it separable?
Find the general solution using any method you like, and find the solution
matching the initial condition y(0)=0.
2. Consider the following differential equation:
dy
t
  ; y (0)  1
dt
y
a) Is this a linear differential equation? Homogeneous? Constant coefficient? If yes,
find the solution to the IVP using any method for solving linear ODEs.
b) Is this a separable differential equation? If yes, find the solution to the IVP using
separation of variables.
3. A dangerous substance known as Chemical X is lethal if its concentration in the air is 100
parts per million by volume (ppmv). The half-life of chemical X is known to be 7 hours.
If I accidentally release a quantity of chemical X in my secret underground lab such that
its initial concentration is 1500 ppmv, how long do I have to wait before I can enter the
lab again?
4. Consider the following differential equation:
y'  y 2  y  6
a) Find the equilibria for this differential equation and assess the stability of each.
For any stable equilibria, determine the basin of attraction.
b) Find the y-coordinates of the inflection points of potential solutions to this
differential equation.
c) Sketch a slope field with relevant solution curves as well.
d) Find the general solution to this differential equation.
5. Consider the following system of differential equations:
dx
 2 x  xy
dt
dy
 3 y  0.5 xy
dt
a) Find and graph the nullclines of this system.
b) What are the equilibria of this system?
c) Classify any equilibria as stable or unstable.
d) Sketch a reasonable solution to this system of differential equations.
6. Find the general solution to the following differential equations:
t
a) y ' y  2e
2
b) y '  1  y
y
1
y ' 
t t  t3
c)
7. A tank with capacity 1000 liters is initially filled with 100 liters of fresh water. Salt
water with a concentration of 6 grams of salt per liter is poured into the tank at a rate of
10 liters per minute. The well stirred mixture flows out of the tank at a rate of 5 liters per
minute. Note that the volume of water in the tank will be increasing until the tank is full.
What will the concentration of salt be at the moment the tank is full?
8. Consider the following system of differential equations:
dx
 5x  4 y  6
dt
dy
 4x  5 y  3
dt
a) Find the equations of, and sketch the nullclines for this system.
b) Find the equilibrium point of this system.
c) Determine whether or not the equilibrium point is stable
9. Classify the following differential equations as linear, separable, both, or neither. If the
differential equation is linear, further classify it as homogeneous or non-homogeneous,
and constant coefficient or variable coefficient. If the differential equation is either linear
or separable, then find the general solution.
dy
 2 y  3e t
dt
a)
dy
 cos 2 ( y ) sin( t )
b) dt
dy
 y2  t2
c) dt
dy
 t2y
d) dt
Solutions will be posted at www.clas.ucsb.edu/staff/vince