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Unformatted text preview: The unknown constants c 1 and c 2 can be determined by the initial condi tions x ( t ) = x and ˙ x ( t ) = u . Let us give some examples of this method. Example 1. Find the solution of the given initial value problem ¨ x + ˙ x 2 x = 0 , x (0) = 2 , ˙ x (0) = 1 and describe its behavior as t increases. We assume that x = e rt and it follows that r must be a root of the charac teristic equation r 2 + r 2 = ( r 1)( r + 2) = 0 . Thus the possible values of r are r 1 = 1 and r 2 = 2 . Therefore, the general solution to the ode is x ( t ) = c 1 e t + c 2 e 2 t . By differentiation we obtain ˙ x ( t ) = c 1 e t 2 c 2 e 2 t . On using the initial conditions, it follows c 1 + c 2 = 2 , c 1 2 c 2 = 1 . By solving above equations simultaneously for c 1 and c 2 we find that c 1 = 1 , c 2 = 1 . Finally, our unique solution that satisfies the ode and the initial conditions is x ( t ) = e t + e 2 t ....
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 Spring '09
 T.Qian

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