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SolutionsODE

# SolutionsODE - Solutions of Ordinary Differential Equations...

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Solutions of Ordinary Differential Equations Consider the n-th order ODE F(x,y, ! y , !! y ,. ..,y n ( ) ) = 0 (1). Definition : Let g(x) be a real-valued function defined on a interval I, having the n-th 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 real-valued 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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SolutionsODE - Solutions of Ordinary Differential Equations...

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