Solutions of Ordinary Differential Equations
Consider the nth order ODE
F(x,y,
!
y ,
!!
y ,.
..,y
n
( )
)
=
0
(1).
Definition
: Let g(x) be a realvalued function defined on a interval I, having the nth
derivative for all x in I.
g(x) is called an
explicit solution
of the equation (1) on the interval I, if:
1)
F(x,g(x),
!
g (x),
!!
g (x),.
..,g
n
(
)
(x))
is defined for all x in I
2)
F(x,g(x),
!
g (x),
!!
g (x),.
..,g
n
( )
(x))
=
0
for all x in I
Example
:
1) Verify that the function g(x) = e
2x
is an explicit solution of the equation
F(x,y,y’,y’’) = y’’ + y’ – 6y = 0
We have g’(x) = 2e
2x
and g’’(x) = 4e
2x
, when we substitute in the equation, we get
F(x, g(x), g’(x), g’’(x))= 4e
2x
+ 2e
2x
– 6 e
2x
= 0 and defined for all x real.
2) Verify that the function h(x) =
x
2
3
defined on the interval (0, 3) is an explicit solution
of the equation F(x,y,y’) = 3xy’ – 2y = 0.
Since g’(x) =
2
3
x
!
1
3
is defined on the interval (0, 3) and
F(x, g(x), g’(x)) =
3x
2
3
x
!
1
3
!
2x
2
3
=
2x
2
3
!
2x
2
3
=
0
and defined for all x in (0, 3)
Definition
: A relation H(x,y) = 0 is called an
implicit solution
of the ODE (1) if this
relation produces at least one realvalued function g(x) defined on the interval I, such that
g(x) is an explicit solution of (1) on I.
Example
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 Fall '08
 STAFF
 Derivative, Ode, dx

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