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CAUCHY-EULER DIFFERENTIAL EQUATIONS - Copy (2)

CAUCHY-EULER DIFFERENTIAL EQUATIONS - Copy (2) - A Alaca...

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A. Alaca MATH 1005 Winter 2010 1 MATH 1005 WINTER 2010 LECTURE SLIDES Prepared by Ay¸ se Alaca Last modified: February 8, 2010 These Slides replace neither the Text Book nor the Lectures CAUCHY-EULER DIFFERENTIAL EQUATIONS
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A. Alaca MATH 1005 Winter 2010 2 CAUCHY-EULER DIFFERENTIAL EQUATIONS Any differential equation of the form a n x n y ( n ) + a n 1 x n 1 y ( n 1) + · · · + a 1 x y + a 0 y = g ( x ) , where a n , a n 1 , a 1 , a 0 are constants, is said to be a Cauchy-Euler equation. In standard form of the nth-order linera differential equation: a n y ( n ) + a n 1 1 x y ( n 1) + · · · + a 1 1 x n 1 y + a 0 1 x n y = g ( x ) x n . Note: We will find the general solution on the interval (0 , ). The solution on ( −∞ , 0) can be obtained by substituting t = x into the differential equation. Method of Solution: To solve ax 2 y ′′ + bxy + cy = g ( x ) , first we will solve ax 2 y ′′ + bxy + cy = 0 . Then by using the method of variation of parameters, we will solve ax 2 y ′′ + bxy + cy = g ( x ) = y ′′ + b ax y + c ax 2 y = g ( x ) ax 2 . ax 2 y ′′ + bxy + cy = 0: We look for a solution of the form y = x m , where m is to be determined. y = x m , y = mx m 1 , y ′′ = m ( m 1) x m 2 .
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