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daktielti 2022-07-07 Answered
Transform the differential equation
{ u ( t ) = sin ( u ( t ) ) u 2 ( t ) , t > 0 , u ( i ) = u i for  i { 0 , 1 , 2 }
into an equivalent system of first order ordinary differential equations.
I know how to carry out this procedure for linear systems but I am stuck because of the sin ( u ( t ) ) term.
For far I tried setting x :≡ u and y :≡ u and then substituting into the equation to obtain multiple differential equations in x and y but don't know how to simplify them to obtain the desired result:
{ y ( t ) = x ( t ) , x ( t ) = sin ( x ( t ) ) u 2 ( t ) , y ( t ) = sin ( y ( t ) ) u 2 ( t ) .
For context: This is the first part of a two part question where in the second part one is asked to find the iterative rule for Eulers method.
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Answers (1)

Kaylie Mcdonald
Answered 2022-07-08 Author has 19 answers
Given the equation
(1) u ( t ) = sin ( u ( t ) ) ( u ( t ) ) 2 ,
we set
(2) v ( t ) = u ( t ) ,
and
(3) w ( t ) = v ( t ) = u ( t ) ;
so that
(4) w ( t ) = v ( t ) = u ( t ) ;
then the equation (1) may be written
(5) w ( t ) = sin ( w ( t ) ) ( u ( t ) ) 2 ,
which together with
(6) u ( t ) = v ( t ) ,
(7) v ( t ) = w ( t ) ,
becomes a first-order system in the three variables u, v, w.
We may in fact "unravel" (5)-(7) to recover (1), simply by substituting (3) and (4) back into (5).
Thus, equation (1) is equivalent to the system (5)-(7).

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