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Unformatted text preview: Linear n-th Order Differential Equations General Form; Properties A linear n-th 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 non-honogeneous 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
- Equations, Quadratic equation, Complex number