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Unformatted text preview: Differential Equations Texas A&M University Math 308, Spring 2008 c circlecopyrt F. Dos Reis Section 5.3 Definition: A system of m differentiable equations in m unknown functions x 1 ( t ) , ,x m ( t ) is in the normal form if it is expressed as x 1 ( t ) = f 1 ( t,x 1 , ,x m ) x 2 ( t ) = f 2 ( t,x 1 , ,x m ) . . . x m ( t ) = f m ( t,x 1 , ,x m ) Exercise 1. Are those systems in the normal form? braceleftbigg x ( t ) = y ( t ) + 3 x ( t )- 5 y ( t ) y ( t ) + x ( t ) = 8 x ( t 0 + y ( t ) braceleftbigg x = 11 x- 2 y y = 5 x + 3 y Exercise 2. Given the initial value problem y + 3 y - 5 y = e t y (0) = 1 ,y (0) = 5 . Convert the initial value problem into a system of initial value problems in the normal form. Recall - Eulers Method formulae for 1 differential equation: Given a differential equation y = f ( x,y ), the initial condition y ( a ) = b and, the step size...
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