306T
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 28-32. See page 34, prob. 35. See page 4-8. See page 16, prob. # 8.
See pages 43-46. See page 35, prob. 37. 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 two examples (applications) 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 two examples (applications) 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.
(e)-[4 pts.]: Choose one of the examples you gave as your answer
for part (d), and set ye= given #. Find the solution of the differential
equation that satisfies the initial condition y(0) = y0 = given #.
Determine the value of the constant of proportionality in the differential
equation if y(t1) = y1 = given # and t1 = 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 a given initial
condition y(0) = y0.
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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) = 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.
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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) = y0.
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 y1 to y(1) if the step size is h1, and that y2 is the Euler
approximation to y(1) if the step size is h2 where h2 < h1 . What reasonable
conjecture can you make about the accuracy of y1 if y1 and y2 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
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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.
Problem # 7: Solve an initial value problem for a Bernoulli Equation
Solve an initial value problem of the form
y '  g (t ) y  r (t ) y n (n  0,1)
y (0)  y0
(See the supplement on Bernoulli equations on the home page)