306FF TEST #1 - Study Guide
Do Any Four Problems- Do One or Two Additional Problems for Extra Credit
Problem #1 Modeling with linear 1st order o.d.e.'s
(See pages 29-31 and 42. See page 34. See page 4. See page 16, prob. # 8. See pages 43, 44, 59-60.
See pages 27-29. Also consider the o.d.e. mv'(t) = kv + mg (k > 0, m=mass > 0, g=gravitational constant > 0) for the velocity of a particle moving vertically under the influence of gravity against air resistance. In all applications that lead to first order inhomogeneous linear differential equations the equation can be written in the form dy/dt = k(y-ye) . )
(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.
(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 y e
is not equal to zero. Define the unknown function and the relevant constants in the differential equation.
(e)-[4 pts.]: Consider the example you gave as your answer for part (d), and set y e
= given #. Find the solution of the differential equation that satisfies the initial condition y(0) = y
0
= given #.
Determine the value of the constant of proportionality in the differential equation if y(t
1
) = y
1
= given # and t
1
= given # . Sketch the solution.
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Problem #2 Solving a 1st Order Linear O.D.E. with Variable Coefficients (See Section 1.8)
(a)-[20 pts]: Solve the differential equation y' = g(t)y + r(t) subject to the initial
condition y(0) = y
0
.
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Problem #3 Separable Equations (See Section 1.2)
(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) = y
0
.
(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) = y
0
.
(d)-[5 pts.]: Find an implicit solution of the following initial value problem:
y' = g(t)h(y), y(0) = y
0
.
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Problem #4 Finding Approximate Solutions O.D.E.'s by Euler's method (See Section 1.4)
(a)-[10 pts.]: Consider the initial value problem
y' = f(t, y), y(0) = y
0
.
Use Euler's method with step size h to approximate y(ti ), i = 1, 2, 3, ..., n.
(b)-[5pts.]: The solution of the given initial value problem is graphed on the
figure below. Draw a polygonal arc connecting the points (ti , yi), i = 1, 2, 3, ..., n,
on this graph where yi (i = 1, 2, 3, ..., n) are the approximations to y(ti ) ( i = 1, 2, 3, ..., n)
found by Euler's method.
(c)-[5 pts.]: Assume that y(t) is the exact solution of an initial value problem or a certain 1st order ordinary differential equation. Suppose that Euler's method yields the approximation y
1
to y(1) if the step size is h
1 approximation to y(1) if the step size is h
can you make about the accuracy of y
1
if y
2
where h
2
< h
1
and y
2
, and that y
2
is the Euler
1
. What reasonable conjecture
agree to m decimal places ?
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Problem #5: Sketching Solutions of Autonomous O.D.Ee.'s (See Section 1.3 and 1.6)
(a)-[5 pts.]: Sketch the graph of z = f(y).
Figure 1
(b)-[3 pts.]: Find the constant solutions of the o.d.e. y ' = f(y) and sketch them on Figure 2.
(c)-[2 pts.] Sketch the phase line for the given o.d.e on Figure 2.
(c)-(f)-[10 pts.]: Sketch non-constant solutions between each pair of constant solutions of y ' = f(y) on Figure 2 .
In answering (c)-(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.
Figure 2

Problem # 6: Sketching solutions of autonomous o.d.e.'s (See Section 1.3 and 1.6 of Text #1)
Given the graph of f(y)
(a)-[5 pts.] - Locate and classify the equilibrium points of the o.d.e. dy/dt = f(y).
(b)-[4 pts. ]Draw the phase line for the given o.d.e.
(c)-[3 pts.] -Sketch the equilibrium solutions of dy/dt = f(y) in the t-y plane.
(d)-[8 pts.] -Sketch a non-constant solution of dy/dt = f(y) that lies in each of the horizontal strips defined by the constant solutions (equilibrium solutions) in the t-y 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 .