306D
TEST #1 - Study Guide
Do Any Three Problems. Do One, Two, Three or Four Additional Problems for Extra Credit
Problem #1:
Modeling with linear 1st order o.d.e.'s
(a)-[4 pts.]: Suppose y(t) represents a certain observable quantity. Express the following
statement as a 1st order ordinary differential equation. The rate of change of y(t) with
respect to t is proportional to y(t).
(b)-[4 pts.]: Give an example (application) of the type discussed in (a). Define the unknown
function and the relevant constants in the differential equation.
Radio-active decay: See probs. #23-25, pp. 40-41.
Personal finance: Read section 3.3. See prob. #1, p 149.
Population growth: Read section 3.1. See probs. #1-4, p. 134.
(Note that all of the above applications lead to first order homogeneous
linear differential equations the equation can be written in the form
dy/dt = ky where k is a positive or negative constant.)
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(c)-[4 pts.] Suppose y(t) represents a certain observable quantity. Express the following
statement as a 1st order ordinary differential equation. The rate of change of y(t) with respect
to t is proportional to the difference of y(t) and a non-zero constant ye.
(The constant ye is a solution of the differential equation.)
(d)-[4 pts.]: Give an example (application) of the type discussed in (c) where ye is not
equal to zero. Define the unknown function and the relevant constants in the differential
equation.
Newton’s law of cooling: See prob. # 35 on p. 42.
Newton’s 2nd law of motion: Consider the o.d.e. mv'(t) = - rv + mg
(r > 0, m=mass > 0, g=gravitational constant > 0) for the velocity of a
particle moving vertically under the influence of gravity, against air resistance.
Read Linear motion and Ex. 3.8, pp. 46-48, See prob. #10 on p. 51.
Mixing Problems: Read section 2.5. See probs. # 1 & 3 on p. 70.
Personal finance: Read section 3.3. See prob. #3 & 5, pp. 149-150
Electric Circuits: Read section 3.4. See Example 4.6 on p. 154.
See probs. #13 & 14, p. 156.
(Note that in all of the above applications the governing differential equation
can be written in the form dy/dt = k(y-ye) where k is a positive or negative constant.)
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(e)-[4 pts.]: Find the solution of the initial value problem dy/dt = k(y-ye), y(0) = y0
where ye and y0 are given numerical constants. Determine the value of k if y(t1) = given #,
and t1 = given positive #. Sketch the solution.
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Problem #2
Separable Equations
(a)-[5 pts.]: Find an implicit general solution of the ordinary differential equation y' = g(t)h(y).
(b)-[5 pts.]: Determine the value of the arbitrary constant in the implicit
general solution that corresponds to the initial condition y(0) = 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(0) = y0.
(d)-[5 pts.]: Find an implicit solution of the following initial value problem: y' = g(t)h(y), y(0) = y0.
(Read Section 2.2, especially the examples. See probs. 13-22 on pg. 40.)
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Problem #3 Solving a 1st Order Linear O.D.E. with Variable Coefficients
(a)-[20 pts]: Solve the differential equation y' = g(t)y + r(t) subject to the initial
condition y(0) = y0.
(Read section 2.4. See Examples 4.6, 4.17, 4.22. See probs. 18-21 on p. 63.)
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306D
TEST #1 - Study Guide
Problem #4:
Solve a Bernoulli Equation
Solve the following initial value problem:
x '  a(t ) y  f (t ) x n (n  0, 1), x(0)  x0
where a(t ) and f (t ) are given funtions of t and n, x0 are given numbers.
(a)-[4pts.]- Reduce the given Bernoulli equation to a first order linear equation for z where x  z1/(1 n ) .
(b)-[12pts.]-Solve the linear equation obtained in part (a) subject to the initial condition z (0)  x0 (1 n )
(c)-[4 pts.]-Use the result of part (b) to find the solution of the original problem.
(Read Prob. #22 on page 63. Do Probs. 23 and 26 on page 63.)
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Problem #5:
Solve a Mixing Problem
[20 pts.]- Solve a mixing problem similar to Example 5.3 on page 69.
(Do problems 12 and 13 on page 72.)
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Problem #6:
Sketching Solutions of Autonomous O.D.Ee.'s
(a)-[5 pts.]- Sketch the graph of z  f ( x) in the x  z plane, where f ( x) is a given
quadratic polynomial in x.
(b)-[3 pts.]- Find the constant solutions of the o.d.e. x '  f ( x) , and sketch them in
the t  x plane.
(c)-[2 pts.]- Draw the phase line for the given o.d.e .
(d)-(f)-[10 pts.]- Sketch non-constant solutions between each pair of constant solutions of x '  f ( x) .
In answering (d)-(f) indicate whether the solution is an increasing or decreasing function of t.
Also in each case indicate where the solution is concave up and where it is concave down.
(Read Example 9.6, pp.111-112. Read “The Phase Line” pp. 112-114. Read Example 9.9, pp. 114-115.
Do probs. 7-10, pp. 117-118. Do probs. 17, 19, 25, 26 pp. 119-129.)
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Problem # 7: Sketching solutions of autonomous o.d.e.'s
Given the graph of f ( x) :
(a)-[5 pts.]- Locate and classify the equilibrium points of the o.d.e. x '  f ( x) .
(b)-[4 pts. ]-Draw the phase line for the given o.d.e.
(c)-[3 pts.]-Sketch the equilibrium solutions of x '  f ( x) in the t  x plane.
(d)-[8 pts.]-Sketch a non-constant solution of x '  f ( x) that lies in each of the horizontal
strips defined by the constant solutions (equilibrium solutions) in the t  x plane. Indicate
where each solution is increasing and where it is decreasing. Indicate where each solution
is concave up and where it is concave down.
(Read Example 9.6, pp.111-112. Read “The Phase Line” pp. 112-114. Read “Stability”, pp. 115-116.
Read Examples 9.11 & 9.12 on page 116. Read Example 9.9, pp. 114-115. Do probs. 7-10, pp. 117-118.
Do probs. 17, 19, 25, 26 pp. 119-129.)
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