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Unformatted text preview: A. Alaca MATH 1005 Winter 2010 2 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A Second-order linear differential equation has the form P ( x ) y + Q ( x ) y + R ( x ) y = G ( x ) ( ) where P,Q,R and G are continuous functions. If G ( x ) = 0 for all x , then ( ) is called homogeneous linear equation. If G ( x ) negationslash = 0, then then ( ) is called nonhomogeneous linear equation. First we will study the homogeneous case. Theorem: If y 1 ( x ) and y 2 ( x ) are both solutions of the linear homogeneous equation P ( x ) y + Q ( x ) y + R ( x ) y = 0 , then y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) , is also a solution of the equation, where c 1 and c 2 are any constants. y 1 ( x ) and y 2 ( x ) are linearly independent y 1 and y 2 are not constant multiples of each other....
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