Chapter 3 Slides - Second Order Linear Dierential Equations...

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Second Order Linear Differential Equations A second order linear differential equa- tion is an equation which can be writ- ten in the form y ±± + p ( x ) y ± + q ( x ) y = f ( x ) (1) where p, q , and f are continuous functions on some interval I . The functions p and q are called the coefficients of the equation. 1
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The function f is called the forcing function or the nonhomogeneous term .
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“Linear” Set L [ y ]= y ±± + p ( x ) y ± + q ( x ) y . Then, for any two twice differentiable functions y 1 ( x ) and y 2 ( x ), L [ y 1 ( x )+ y 2 ( x )] = L [ y 1 ( x )] + L [ y 2 ( x )] and, for any constant c , L [ cy ( x )] = cL [ y ( x )] . That is, L is a linear differential operator. 2
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Existence and Uniqueness THEOREM Given the second order linear equation (1). Let a be any point on the interval I , and let α and β be any two real numbers. Then the initial-value problem y ±± + p ( x ) y ± + q ( x ) y = f ( x ) , y ( a )= α, y ± ( a β has a unique solution. 3
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Homogeneous/Nonhomogeneous Equations The linear differential equation y ±± + p ( x ) y ± + q ( x ) y = f ( x ) (1) is homogeneous if the function f on the right side is 0 for all x I .I n this case, the equation becomes y ±± + p ( x ) y ± + q ( x ) y =0 . (1) is nonhomogeneous if f is not the zero function on I . 4
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Homogeneous Equations y ±± + p ( x ) y ± + q ( x ) y = 0 (H) where p and q are continuous func- tions on some interval I . Trivial Solution The zero function, y ( x ) = 0 for all x I ,( y 0) is a solution of (H). ( y 0 implies y ± 0 and y ±± 0). The zero solution is called the trivial solution . Any other solution is a nontrivial solution. 5
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Basic Theorems THEOREM 1 If y = y ( x ) is a solu- tion of (H) and if C is any real num- ber, then u ( x )= Cy ( x ) is also a solution of (H). Any constant multiple of a solution of (H) is also a solution of (H) . 6
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THEOREM 2 If y = y 1 ( x ) and y = y 2 ( x ) are any two solutions of (H), then u ( x )= y 1 ( x )+ y 2 ( x ) is also a solution of (H). The sum of any two solutions of (H) is also a solution of (H) . (Some call this property the superposition principle ). 7
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DEFINITION ( Linear Combinations ) Let f = f ( x ) and g = g ( x ) be func- tions deFned on some interval I , and let C 1 and C 2 be real numbers. The expression C 1 f ( x )+ C 2 g ( x ) is called a linear combination of f and g . 8
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THEOREM 3 If y = y 1 ( x ) and y = y 2 ( x ) are any two solutions of (H), and if C 1 and C 2 are any two real numbers, then y ( x )= C 1 y 1 ( x )+ C 2 y 2 ( x ) is also a solution of (H). Any linear combination of solutions of (H) is also a solution of (H) . 9
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NOTE: y ( x )= C 1 y 1 ( x )+ C 2 y 2 x is a two-parameter family which ”looks like“ the general solution. Is it??? Example: y ±± - 1 x y ± - 15 x 2 y =0 a. y 1 ( x x 5 ,y 2 ( x )=3 x 5 y = C 1 x 5 + C 2 (3 x 5 ) b. y 1 ( x x 5 2 ( x x - 3 y = C 1 x 5 + C 2 x - 3 10
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DEFINITION: Wronskian Let y = y 1 ( x ) and y = y 2 ( x ) be solu- tions of (H). The function W deFned by W [ y 1 ,y 2 ]( x )= y 1 ( x ) y ± 2 ( x ) - y 2 ( x ) y ± 1 ( x ) is called the Wronskian of y 1 2 . Determinant notation: W ( x y 1 ( x ) y ± 2 ( x ) - y 2 ( x ) y ± 1 ( x ) = ± ± ± ± ± ± y 1 ( x ) y 2 ( x ) y ± 1 ( x ) y ± 2 ( x ) ± ± ± ± ± ± 11
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THEOREM 4 Let y = y 1 ( x ) and y = y 2 ( x ) be solutions of equation (H), and let W ( x ) be their Wronskian. Ex- actly one of the following holds: (i) W ( x ) = 0 for all x I and y 1 is a constant multiple of y 2 or vv.
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This note was uploaded on 02/09/2011 for the course MATH 3321 taught by Professor Morgan during the Fall '08 term at University of Houston.

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Chapter 3 Slides - Second Order Linear Dierential Equations...

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