Math 310 Q8 Sol - MATH 310, Fall 2011: Differential...

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MATH 310, Fall 2011: Differential Equations Simon Fraser University Quiz 8: Solutions 1. Given that y 1 ( x ) = x is a solution of the differential equation x 2 y 00 + 2 xy 0 - 2 y = 0 ( x > 0) , use the method of reduction of order or Abel’s formula to find a second linearly independent solution y 2 ( x ), and hence write down the general solution. Solution: 1) Using the method of reduction of order : We seek a second solution of the DE of the form y 2 ( x ) = u ( x ) y 1 ( x ) = xu ( x ) . Dif- ferentiating, we have y 0 2 ( x ) = u ( x ) + xu 0 ( x ) and y 00 2 ( x ) = 2 u 0 ( x ) + xu 00 ( x ) . Now we substitute into the DE, to get x 2 y 00 2 + 2 xy 0 2 - 2 y 2 = x 2 (2 u 0 + xu 00 ) + 2 x ( u + xu 0 ) - 2 xu = x 3 u 00 + (2 x 2 + 2 x 2 ) u 0 + (2 x - 2 x ) u = 0 , or x 3 u 00 + 4 x 2 u 0 = 0 , which is a first-order linear (also separable) differential equation for v = u 0 . Writing it in the form v 0 + 4 x v = 0 , the integrating factor is μ ( x ) = e R (4 /x ) dx = x 4 , so that we have x 4 v 0 + 4 x 3 v = d dx ± x 4 v ² = 0 = x 4 v = constant = c, giving v = cx - 4 . Recalling that v = u 0 , this gives u 0 = v = cx - 4 = u ( x ) = - c 3 x - 3 + d = ˜ cx - 3 + d.
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Math 310 Q8 Sol - MATH 310, Fall 2011: Differential...

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