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  3. x 1 thus y 1 1 91 441 6 2 3 1 91 17 2...

x 1 Thus y 1 1 91 441 6 2 3 1 91 17 2...

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Calculus

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x 1 Thus y 1 1 91 441 6 2 3 1 91 17 2 6 3 31 8 2 6 HIGHER ORDER ODE 41 271 41 4 16 As mentioned earlier we associate a first order ODE with a order k ODE and try to get information about the solutions of the higher order ODE in terms of those of the associated first order ODE In particular we are interested in a condition on the function f f t X which will di di ensure existence and uniqueness of the initial value problem for the higher order ODE In this article we explain the theory To begin with we consider the ODE dx dx f 1 x dt 4 Together with the initial condition dx MATHEMATICS di We introduce a new variable y 2 where y ranges in 2 and y y ranging in R Next we define F 1x2xRx R R by putting F t y F t Y Y3 f t y Now the given initial value problem for the order k ODE give rise to the following initial value problem in the first order ODE to W 1 dr F t y 5 the initial condition for it being y t x W W Clearly the above first order ODE is actually the following system of first order ODE 27

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