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Chapter 3 Lecture Notes Complete ppt

Chapter 3 Lecture Notes Complete ppt - Second Order...

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Second Order Differential Equations Implicit Form: ( , , , ) 0 F x y y y ′′ = Explicit Form: ( , , ) y f x y y ′′ = Boundary Condition: 0 0 0 0 ( ) , ( ) y x y y x y = = Existence of Unique Solution: If , , ' f f f y y are continuous in a region R, and ( 29 0 0 0 , , ' P x y y R = , then there exists a region around P where solution is unique. Linear/Nonlinear, Second Order Differential Equation Forms: ( ) (Nonhomogeneous) ( ) ( ) ( ) 0 (Homogeneous) G x P x y Q x y R x y ′′ + + = Or ( ) (Nonhomogeneous) ( ) ( ) 0 (Homogeneous) g x y p x y q x y ′′ + + = , ( ) 0 P x Otherwise, the differential equation is nonlinear. For the existence of a unique solution, assume that the functions , , , , , , are contiunuous in <x< P Q R G p q g α β . Here may go to - , and may go to α β
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Existence of Solution
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Linear Equations: Structure of General Solution ( ) (Nonhomogeneous) ( ) ( ) 0 (Homogeneous) g x y p x y q x y ′′ + + = Let ( ) c y y x = is a solution of the homogeneous equation, and ( ) p y y x = is any solution of the nonhomogeneous equation. Then the ( ) ( ) ( ) c p y x y x y x = + is a solution of the linear, differential equation. Complimentary Solution: ( ) c y y x = Particular Solution: ( ) p y y x = SECOND ORDER, LINEAR DIFFERENTIAL EQUATION
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Operator Form of Linear Solution Define the differential operator 2 2 ( ) ( ) d d L p x q x dx dx = + + , then c y satisfies [ ] 0 c L y = and p y satisfies [ ] ( ) p L y g x = L is a linear operator i.e., 1 2 1 2 [ ] [ ] [ ] , , constants. L y y L y L y α β α β α β + = + Notations of constants : ' c s with or without subscripts are constanst, unless otherwise specified. Principle of Superposition : If 1 2 [ ] 0 [ ] L y L y = = , then 1 1 2 1 1 2 [ ] [ ] [ ] =0 L c y c y c L y c L y + = + If 1 2 , y y are solutions of the homogeneous equation, then 1 1 2 2 c y c y + is also a solution of the homogeneous equation
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Wronskian of Two Functions 1 2 1 2 2 1 ( ) ( , ) W x W y y y y y y = = - Linear Independence Two functions 1 2 and y y are said to be linearly independent if the only solution of 1 1 2 2 1 2 0 is 0 c y c y c c + = = = Note: 1 2 and y y are linearly independent in x α β < < , then 1 2 1 2 2 1 ( ) ( , ) 0 W x W y y y y y y = = - in x α β < < Wronskian of Two Solutions of Homogeneous Equations 1 2 [ ] 0 [ ] L y L y = = in x α β < < ( ) 1 2 1 2 2 1 ( ) ( , ) = x p t dt W x W y y y y y y ce - = = - ( ) W x is either identically zero or is never zero in x α β < <
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Wronskian of Two Solutions of a Linear Second Order Differential Equation (SLDE) 21(158)8
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General Solution of Homogeneous Linear Equation [ ] 0 = L y If 1 2 and y y are two linearly independent solutions 1 2 [ ] 0 [ ] L y L y = = in x α β < < satisfying 0 0 0 0 ( ) , ( ) y x y y x y = = Then any solution i.e., general solution
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