above is a typographical
error.
(a)
(5 pts.) Write down the general solution to the differential equation.
[
WARNING: Be very careful.
This will be graded Right or Wrong!!
]
(b) (5 pt.) What is the order of the differential equation?
The order of the differential equation is 8.
______________________________________________________________________
6. (15 pts.)
(a)
Obtain the differential equation satisfied by the
family of curves defined by the equation (*) below.
(b) Next, write down the differential equation that the
orthogonal trajectories to the family of curves defined by (*) satisfy.
(c) Finally, solve the differential equation of part (b) to
obtain the equation(s) defining the orthogonal trajectories. [These, after
all, are another family of curves.]
(*)
.
(a)
Differentiating (*) with respect to x and then replacing
c
using (*)
yields
a differential equation for the family of curves.
(b)
An ODE for the orthogonal trajectories is now given by
or equivalently,
.
(c)
This little equation is equivalent to one that is linear with x as the
dependent variable and y the independent variable(!):
This linear equation has
μ
= exp(y) as an integrating factor.
Using the
standard recipe, a oneparameter family of solutions is given by
.
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______________________________________________________________________
7.
(10 pts.)
The nonzero function
f
(
x
) = exp(
x)
is a solution to the
homogeneous linear O.D.E.
(*)
(a)
Reduction of order with this solution involves making the substitution
into equation (*) and then letting
w
=
v
′
.
Do this substitution and obtain
in standard form the first order linear homogeneous equation that
w
must
satisfy.
(b)
Finally, obtain an integrating factor,
μ
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 Fall '08
 STAFF
 Vector Space

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