306 G
FINAL EXAM - Study Guide
12/17/07
1
Do Any Six Problems- Do One or Two Additional Problems for Extra Credit
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Problem #1 Solving a 1st Order Linear O.D.E. with Variable Coefficients
(a)-[20 pts]: Solve the differential equation y' + p(t)y = g(t) subject to the initial
condition y(t0) = y0 where p(t) and g(t) are given continuous functions on the interval
I=(, ) and t0  I.
( See Section 2.1, especially Example 1 and Example 2. See Chapter Summary on
pg. 118. Also see prob. # 19 on pg. 47. )
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Problem #2
Separable Equations
(a)-[5 pts.]: Find an implicit general solution of an ordinary differential
equation of the form M(x) + N(y)y' = 0.
(b)-[5 pts.]: Determine the value of the arbitrary constant in the implicit
general solution that corresponds to the initial condition y(t0) = y0.
(c)-[5 pts.]: Use the result of part (b) to find the explicit solution of the given
differential equation that satisfies the initial condition y(t0) = y0.
(See Section 2.2, especially Examples 1-3. See Chapter Summary on pg. 39. Also
see prob. 25, pg 55.)
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Problem #3:
Autonomous Differential Equations
(a)-[5 pts.]-Locate and classify the critical points of the differential equation y' f (y)
in the t-y plane. Assume that f (y) is a polynomial.
(b)-[5 pts.] Sketch the given slope function f (y) .

(c)-[5 pts.]- Sketch
 the phase line for the differential equation given in part (a).
(d)-[5 pts.]- Sketch the equilibrium
 (constant) solutions corresponding to the critical
points found in part (a) in the t-y plane. In each strip of the t-y plane determined by the
equilibrium solutions sketch a non-constant solution of the given differential equation.
Use the results of (a)-(c) to determine whether each of these solutions is increasing or
decreasing. Use the graph of f (y) found in part (b) to determine where each of these
solutions is concave up, and where it is concave down.
(See section 2.5, especially example 1 on page 82, and problem 3 on page 92.) (See
discussion on top
half of page 87. See problem 3 on page 92. Read and understand
problem 7 on page 92. See problems 8-13 on page 92.) (See Fig. 2.5.1 on page 91. See
problem 1-6 on page 91-92.)
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Problem # 4:
Exact Equations and Integrating Factors
(a)-[10 pts.]- Find an integrating factor for a given differential equation of the form
M(x, y)  N(x, y) y' = 0.

 
(b)-[10 pts]- Use the result of part (a) to find a function  (x, y) such that for any constant
c,  (x, y)  c is an implicit solution of the differential equation given in part (a),
(See section 2.6. See Example 4 on page 101.
See problems 25, 27, 29 on page 103.)
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306 G
Problem # 5:
FINAL EXAM - Study Guide
12/17/07
Homogeneous Linear Systems of ODE’s with Constant Coefficients
(a)-[12 pts.]- Find a fundamental pair of vector solutions of the linear system x’ = Ax
a11 a12 
x 
where x =  1 , and A = 
.
x 2 
a 21 a 22 
HINT: Construct a fundamental pair of solutions of the form x  e t v where  is an
v 
eigenvalue of A and v =  1 is a corresponding eigenvector.
v 2 






2
306 G
FINAL EXAM - Study Guide
12/17/07
3
(b)-[8 pts.]- Use the result of part (a) to find the solution of the system given in part (a)
x1 (0) x10 
that satisfies the initial condition x(0)  
  0 .
x2 (0) x2 
(See section 3.3. See Theorem 3.3.1 on page 152 and Theorem 3.3.2 on page 153. See
examples 3 and 4 on pages 153 and 157. See problems 13, 14, 15 on page 162.)
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
Problem 6:
Find a function H(x,y) such that the trajectories of the first order
dx
dy
system
 f (x, y),
 g(x, y) lie on the level curves of H(x,y).
dt
dt
(Section 3.6,especially example 1 on page 189 of sect. 3.6. Also see problems 10, 11
and 12 on page 193. )

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Problem 7:
Given an almost linear first order system (S) of the form
dx
dy
 f (x, y),
 g(x, y) where f (x,y) and g(x,y) are polynomials in
dt
dt
x and y determine the following:


(a)-[5 pts.]- the critical points
 of (S), (b)-[5 pts.] - the corresponding linear
system near each critical point, (c)-[5 pts.] - the eigenvalues of each linear
system.
(d)-[5 pts.]-Use the results of (a)-(c) to classify the critical points of (S).
(See section 7.1 and 7.2, especially Theorem 7.2.2 on page 487 and table 7.2.2 on
page 488. See Example 3 on page 486 and Example 4 on page 487.)
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Problem 8:
See section 6.2. A problem like Example 3 on page 407.
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Problem 9:
See section 6.3. A problem like Example 1 on page 413.
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