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Math 231
Differential Equations
Summer
Test #1
6/12=F
BJB
Directions:
Do all work neatly in the answer booklet provided. Show all of your work. Partial credit is given.
Leave no problem unanswered. Number problems as they are numbered on this paper. Simplify your
answers as much as possible and circle your final answer where appropriate. All problems are worth 10
points except where noted. Label your graphs completely.
1.)
Define: A differential equation and its solution.
2.)
Give a simple example of a first order ordinary linear differential equation, (ODE).
3.)
Find dy/dx, if: 3x sin y – 2y ln x = 4.
4.)
Find f/y, if f(x,y) = x tan y – 2 sec x + exy
(5 points)
Integrate the following problems:
5.)  2 sec2 3x dx
x
6.)  (3 x 2  cos )dx
2
7.)  x ex dx
Classify (but do not solve) each of the following differential equations as an ordinary differential equation
(ODE) or a partial differential equation (PDE). If they are ODE give the order, indicate the independent and
dependent variables and state whether the equation is linear or nonlinear. Give reasons:
(5 points each)
8.)
2xy(5) + x2y + x4y - 3 ln y = x2 – 7x +12
9.)
y
10.)
e 2t s  sin t s  3s  et
f ( x, y )
f ( x, y )
x
x
y
Solve the following differential equations and state what type of problem you are solving; where possible,
give the explicit solution.
11.)
12.)
dy
 (1  y 2 ) tan x, y (0)  0
dx
Using Euler’s method and with step size: h = 0.5 approximate the solution to
dy
 x  y, y (0)  1 from x = 0 to x = 1.5. Show your work. You may use your calculator
dx
to check your work. Plot the graph of these three points on the first set of axes. Plot three
points of the slope field of this differential equation on the second set of axes for the points:
(0, 0), (1, 2) and (2, 3).
(20 points)