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Unformatted text preview: Section 1.2: Solutions Deﬁnition 1 f is an explicit solution of the diﬀerential equation
,..., n] = 0
if the identity F [x, f (x), f (x), . . . , f (n) (x)] is deﬁned for all x ∈
I , and the identity = 0 for all x ∈ I . Meaning that the substitution of
f (x) and its various derivatives for y and corresponding derivatives
reduces F to an identity on I .
F [x, y, Example 1
f (x) = 2sinx + 3cosx
is an explicit solution of
f is deﬁned and has a second derivative
f (x) = 2cosx − 3sinx
f (x) = −2sinx − 3cosx
Substitute into the diﬀerential equation
(−2sinx − 3cosx) + (2sinx + 3cosx) = 0
which holds for all real x. So, the function f deﬁned by f (x) =
2sinx +3cosx is an explicit solution of the diﬀerential equation dxy +
y = 0 for all real x.
Deﬁnition 2 A relation g (x, y ) = 0 is an implicit solution of 1 dy
F [x, y, , . . . , n ] = 0
if this relation deﬁnes at least one real function f of the variable
x on an interval I such that this function is an explicit solution on
Example 2 The relation x2 + y 2 − 25 = 0 is an implicit solution of
the diﬀ eq
dx x2 + y 2 − 25 = 0
This relation deﬁnes two real functions:
f1 (x) = 25 − x2
f2 (x) = − 25 − x2
Both of these are explicit solutions of the diﬀ eq on I . So, the
relation x2 + y 2 − 25 = 0 is an implicit solution of the diﬀ eq.
x2 + y 2 + 25 = 0
2x + 2y dy
y After substitution
x + y (− ) = 0
The relation formally satisﬁes the diﬀ eq. A formal solution has
appearance of a solution. 2 ...
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- Spring '11