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Unformatted text preview: MA 36600 FINAL REVIEW Chapter 7 7.1: Introduction. A system of first order equations is a collection of initial value problems dx k dt = G k ( t, x 1 , x 2 , ..., x n ) , x k ( t ) = x k ; k = 1 , 2 , ..., m. We say that such a system is a system of first order linear equations if we can write G k ( t, x 1 , x 2 , ..., x n ) = p k 1 ( t ) x 1 + p k 2 ( t ) x 2 + + p kn ( t ) x n + g k ( t ); k = 1 , 2 , ..., m. If at least one of the G k ( t,x 1 ,x 2 ,...,x n ) is not in the form above, we call the system nonlinear . We call a linear system homogeneous if each g k ( t ) is the zero function. Otherwise, we call the system nonhomogeneous . We can always express an n th order linear equation as a system of first order equations: In general, an n th order equation is an ordinary differential equation in the form y ( n ) = G t, y, y (1) , ..., y ( n 1) where the notation y ( k ) denotes the k th derivative; and we have the initial conditions y ( t ) = y , y (1) ( t ) = y (1) , ... y ( n 1) ( t ) = y ( n 1) . If we make the substitutions x 1 = y x 2 = y (1) . . . x n = y ( n 1) = x 1 = x 2 . . . x n 1 = x n x n = G ( t, x 1 , ..., x n ) where x 1 = y x 2 = y (1) . . . x n = y ( n 1) Assume that there are as many equations as variables i.e., that m = n . Say that there is a region R = ( t, x 1 , ..., x n ) R n +1 < t < k < x k < k for k = 1 , 2 , ..., n such that i. Each G k ( t,x 1 ,x 2 ,...,x n ) is continuous on R . (There are m functions to consider here.) ii. Each G i x j is continuous on R . (There are m n functions to consider.) iii. ( t , x 1 , ..., x n ) R . Then there exists a subregion R R such that the system of differential equations has a unique solution. This is known as the Existence and Uniqueness Theorem for Nonlinear Systems . Consider a system of first order linear equations dx k dt = p k 1 ( t ) x 1 + p k 2 ( t ) x 2 + + p kn ( t ) x n + g k ( t ) , x k ( t ) = x k ; k = 1 , 2 , ..., m. Again, for simplicity assume that m = n . Say that there is an interval I = t R < t < 1 2 MA 36600 FINAL REVIEW such that i. Each g k ( t ) is continuous on I . (There are m functions to consider.) ii. Each p ij ( t ) is continuous on I . (There are m n functions to consider.) iii. t I . Then there exists a unique solution on the interval I . This is known as the Existence and Uniqueness Theorem for Linear Systems . Consider a LRC Circuit i.e., an electrical circuit which contains three objects: An Inductor (which behaves like a mass) V inductor dI inductor dt = V inductor = L dI inductor dt ....
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 Spring '09
 MA
 Equations

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