m2k_dfq_scornh

m2k_dfq_scornh - Inhomogeneous Linear Second Order...

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Inhomogeneous Linear Second Order Diferential Equations Properties of Solutions; the General Solution We turn our attention now to linear inhomogeneous second order equations , which we may assume written in the form d 2 y dx 2 + p ( x ) dy dx + q ( x ) y = g ( x ) , where p ( x ) , q ( x ) and g ( x ) are known continuous, or piecewise contin- uous, functions deFned on some interval a < x < b . Since we are treating the inhomogenous case, the assumption is that g ( x ) is not identically equal to 0 on that interval. We will begin by listing some properties of solutions. 1) If y 1 ( x ) is a solution of the inhomogeneous equation above and z ( x ) is a solution of the homogeneous equation d 2 z dx 2 + p ( x ) dz dx + q ( x ) z = 0 , then y 2 ( x ) = y 1 ( x ) + z ( x ) is also a solution of the inhomogeneous equation. 2) If y 1 ( x ) and y 2 ( x ) are solutions of the inhomogeneous equation just indicated, then the di±erence, z ( x ) = y 2 ( x ) - y 1 ( x ), is a solution of the corresponding homogeneous equation. Equivalently, there is a solution z ( x ) of the homogeneous equation 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 solutions of the homogeneous 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 2 y dx 2 + p ( x ) dy dx + q ( x ) y = g ( x ) = α g 1 ( x ) + β g 2 ( x ) . 1
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4) Suppose the general solution of the homogeneous equation takes the form z ( x, c 1 , c 2 ) = c 1 z 1 ( x ) + c 2 z 2 ( x ) , where c 1 and c 2 are arbitrary constants and z 1 ( x ) and z 2 ( x ) are two so- lutions of the homogeneous equation. Let y p ( x ) be a particular solution (i.e., any particular solution) of the inhomogeneous equation. Then the general solution of the inhomogenous equation is given by y ( x, c 1 , c 2 ) = c 1 z 1 ( x ) + c 2 z 2 ( x ) + y p ( x ) . The Frst three of these properties are more or less evident from the linear form of the equation and require no special proof. The fourth property does merit some demonstration. Proof of 4) . Suppose for some x 0 in the interval a < x < b and given values y 0 and y 1 we are required to satisfy the conditions y ( x 0 ) = y 0 , y 0 ( x 0 ) = y 1 . Then, using the indicated solution form, we need to choose c 1 and c 2 such that c 1 z 1 ( x 0 ) + c 2 z 2 ( x 0 ) + y p ( x 0 ) = y 0 , c 1 z 1 0 ( x 0 ) + c 2 z 2 0 ( x 0 ) + y p 0 ( x 0 ) = y 1 .
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m2k_dfq_scornh - Inhomogeneous Linear Second Order...

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