Solving this and substituting back results in a 1parameter family of
implicit solutions:
.
Getting an explicit solution now is cheap thrills.
______________________________________________________________________
Bonus Noise
:
The substitution
v
=
y
′
converts the ODE
into the first order linear equation
(*)
Using the integrating factor
µ
= e
x
leads to the explicit solution to (*)
.
Finally,
.
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______________________________________________________________________
(c)
Linear
as written.
Near
θ
= 0, an integrating factor is easy to come by:
for
θ
ε
(
π
/2,
π
/2). Multiplying both sides of the DE by
µ
results in the
following derivative equation:
By integrating, we have
Applying the initial condition, it follows that C = 4. Hence, an explicit
solution near
θ
= 0 is given by
(d)
After minimal algebra, this may be seen as either
exact
with an easy
solution, or
homogeneous
, of degree 1, with a messier solution.
As Exact
: The usual prestidigitation leads to
A 1parameter family of solutions:
As Homogeneous:
The degree of homogeneity is 1.
Thus, write the
equation in the form of
dy
/
dx
=
g
(
y
/
x
) by doing suitable algebra carefully.
After setting
y
=
vx
, substituting, and doing a bit more algebra, you will
end up looking at the separable equation
.
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 Fall '08
 STAFF
 Differential Equations, Derivative, Ode

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