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Unformatted text preview: Systems of Differential Equations Systems; Solutions An n-dimensional system of differential equations involves an unknown n-vector function Y ( t ) = y 1 ( t ) y 2 ( t ) . . . y n ( t ) . Such a vector function is differentiable just in case each (scalar) component function y k ( T ) is differentiable, k = 1 , 2 , ..., n and the derivative is Y prime ( t ) = y prime 1 ( t ) y prime 2 ( t ) . . . y prime n ( t ) . A first order system of differential equations is written in the form Y prime ( t ) = F ( t, Y ( t )) , by which we mean y prime 1 ( t ) y prime 2 ( t ) . . . y prime n ( t ) = f 1 ( t, y 1 ( t ) , y 2 ( t ) , ..., y n ( t )) f 2 ( t, y 1 ( t ) , y 2 ( t ) , ..., y n ( t )) . . . f n ( t, y 1 ( t ) , y 2 ( t ) , ..., y n ( t )) . A little later we will see that virtually all systems, of whatever order, can be re- expressed as first order systems so these are all we need to consider at this point. Example 1 The system of equations parenleftbigg y prime 1 ( t ) y prime 2 ( t ) parenrightbigg = parenleftBigg 2 y 1 ( t ) y 2 ( t ) y 1 ( t ) 2 + y 2 ( t ) 2 parenrightBigg constitutes a 2 dimensional system of nonlinear differential equations; nonlinear be- cause the component functions on the right hand side are not linear in y 1 ( t ) and y 2 ( t ). A solution of a first order system such as this is a differentiable vector function Y ( t ) which, substituted into the system equations, yields a vector identity. In the case of nonlinear systems such as this one it is frequently quite difficult, or even impossible, 1 to find closed form expressions for solutions. However, we can find solutions for this particular example. Let w ( t ) = y 2 ( t ) + y 1 ( t ) , z ( t ) = y 2 ( t )- y 1 ( t ) . Adding the two equations of the system and then subtracting the first equation from the second, we obtain w prime ( t ) = y prime 2 ( t ) + y prime 1 ( t ) = y 1 ( t ) 2 + 2 y 1 ( t ) y 2 ( t ) + y 2 ( t ) 2 = parenleftbigg y 2 ( t ) + y 1 ( t ) parenrightbigg 2 = w ( t ) 2 , z prime ( t ) = y prime 2 ( t )- y prime 1 ( t ) = y 1 ( t ) 2- 2 y 1 ( t ) y 2 ( t ) + y 2 ( t ) 2 = parenleftbigg y 2 ( t )- y 1 ( t ) parenrightbigg 2 = z ( t ) 2 .....
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