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Unformatted text preview: Linear nth Order Differential Equations General Form; Properties A linear nth order differential equation is an equation of the form d n y dx n + p 1 ( x ) d n 1 y dx n 1 + ... + p n 1 ( x ) dy dx + p n ( x ) y = g ( x ) , where the p k ( x ) , k = 1 , 2 , ..., n and g ( x ) are known continuous, or at least piecewise continuous, functions defined on some in terval a < x < b (which may be (∞ , ∞ ). We make the usual distinction between homogeneous and nonhonogeneous equa tions, according as g ( x ) is, or is not, identically equal to zero on ( a, b ), respectively. A solution is an n times differentiable function, y ( x ), defined on ( a, b ), which, on substitution into the equation, reduces the equation to an identity. The general prop erties are essentially the same as we have already listed for the inhomogeneous equation in the second order case: • 1) If y 1 ( x ) is a solution of the inhomogeneous equation and z ( x ) is a solution of the corresponding homogeneous equation d n z dx n + p 1 ( x ) d n 1 z dx n 1 + ... + p n 1 ( x ) dz dx + p n ( x ) z = 0 , then y 2 ( x ) = y 1 ( x ) + z ( x ) is also a solution of the inhomoge neous equation. • 2) If y 1 ( x ) and y 2 ( x ) are solutions of the inhomogeneous equation just indicated, then the difference, z ( x ) = y 2 ( x ) y 1 ( x ), is a solution of the corresponding homogeneous equation. 1 Equivalently, there is a solution z ( x ) of the homogeneous equa tion such that y 2 ( x ) = y 1 ( x ) + z ( x ). • 3) If g ( x ) = α g 1 ( x ) + β g 2 ( x ) and y 1 ( x ) and y 2 ( x ) are so lutions of the inhomogeneous equation with g ( x ) replaced by g 1 ( x ) and g 2 ( x ), respectively, then y ( x ) = α y 1 ( x ) + β y 2 ( x ) is a solution of d n y dx n + p 1 ( x ) d n 1 y dx n 1 + ... + p n 1 ( x ) dy dx + p n ( x ) y = g ( x ) = α g 1 ( x ) + β g 2 ( x ) . • 4) The general solution of the homogeneous equation takes the form z ( x, c 1 , c 2 , ..., c n ) = c 1 z 1 ( x ) + c 2 z 2 ( x ) + ... + c n z n ( x ) , where c 1 , c 2 , ..., c n are arbitrary constants and z 1 ( x ) , z 2 ( x ) , ..., z n ( x ) are n solutions of the homogeneous equation forming a funda mental set of solutions  a concept which will explain a little later. Letting y p ( x ) be a particular solution (i.e., any solution) of the inhomogeneous equation, the general solution of the in homogenous equation is given by y ( x, c 1 , c 2 , ..., c n ) = c 1 z 1 ( x ) + c 2 z 2 ( x ) + ... + c n z n ( x ) + y p ( x ) ....
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 Spring '10
 ROBINSON
 Equations, Quadratic equation, Complex number

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