Section 1.2 Notes - Section 1.2: Solutions Definition 1 f...

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Unformatted text preview: Section 1.2: Solutions Definition 1 f is an explicit solution of the differential equation dy dn y ,..., n] = 0 dx dx 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 d2 y +y =0 dx2 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) = d2 2sinx +3cosx is an explicit solution of the differential equation dxy + 2 y = 0 for all real x. Definition 2 A relation g (x, y ) = 0 is an implicit solution of 1 dy dn y F [x, y, , . . . , n ] = 0 dx dx 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 this interval. Example 2 The relation x2 + y 2 − 25 = 0 is an implicit solution of the diff eq x+y dy =0 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. Example 3 x2 + y 2 + 25 = 0 x+y 2x + 2y dy =0 dx dy dy x =0⇒ =− dx dx y After substitution x x + y (− ) = 0 y The relation formally satisfies the diff eq. A formal solution has appearance of a solution. 2 ...
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Section 1.2 Notes - Section 1.2: Solutions Definition 1 f...

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