1 Why do we need the restriction t t h a 2 What is the significance of the

1 why do we need the restriction t t h a 2 what is

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1 Why do we need the restriction | t - t 0 | ≤ h ≤ | a | ? 2 What is the significance of the conditions f and ∂f/∂y being continuous in R ? Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (21/23)
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Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Examples 2 Consider the differential equation y 0 = y 2 , with y (0) = 1 Note that f ( y ) = y 2 and ∂f/∂y = 2 y , which are continuous in any rectangle R This is a separable equation, so Z y - 2 dy = Z dt = t + C or - 1 y ( t ) = t + C The solution to the IVP is y ( t ) = 1 1 - t , which clearly becomes undefined at t = 1. The interval of existence does not match the interval of continuity for f ( t, y ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (22/23)
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Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Examples 3 Consider the differential equation y 0 = y 2 / 3 , with y (0) = 0 Note that f ( y ) = y 2 / 3 is continuous in any rectangle centered at ( t 0 , y 0 ) = (0 , 0), while ∂f/∂y = 2 3 y - 1 / 3 , which is NOT continuous in any rectangle R near (0 , 0) This is a separable equation, so Z y - 2 / 3 dy = Z dt = t + C or 3 y ( t ) 1 / 3 = t + C One solution to the IVP is y ( t ) = t 3 27 , which satisfies the IVP. However, it is easy to see that y ( t ) 0 is a solution, so solutions are NOT unique Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (23/23)
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