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Unformatted text preview: Systems of Differential Equations Systems; Solutions An ndimensional system of differential equations involves an unknown nvector 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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 Spring '10
 ROBINSON
 Differential Equations, Equations, Scalar, Trigraph, dt

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