HW1 - Math 293 Fall 2005 Homework Solution 1 Section 1.1...

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Math 293 - Fall 2005 Homework Solution 1 - Section 1.1 Alexandre ROCH September 1, 2005 Problem. 3. y 00 + 4 y = 0; y 1 = cos 2 x y 2 = sin 2 x y 0 1 = - 2 sin 2 x y 00 1 = - 4 cos 2 x = ( - 4 cos 2 x ) + 4(cos 2 x ) = 0 y 0 2 = 2 cos 2 x y 00 2 = - 4 sin 2 x = ( - 4 cos 2 x ) + 4(cos 2 x ) = 0 Problem. 6. y 00 + 4 y 0 + 4 y = 0; y 1 = e - 2 x y 2 = xe - 2 x y 0 1 = - 2 e - 2 x y 00 1 = 4 e - 2 x = (4 e - 2 x ) + 4( - 2 e - 2 x ) + 4( e - 2 x ) = 0 y 0 2 = e - 2 x - 2 xe - 2 x y 00 2 = - 4 e - 2 x +4 xe - 2 x = ( - 4 e - 2 x +4 xe - 2 x )+4( e - 2 x - 2 xe - 2 x )+4( xe - 2 x ) = 0 Problem. 7. y 00 - 2 y 0 + 2 y = 0; y 1 = e x cos x y 2 = e x sin x y 0 1 = e x cos x - e x sin x y 00 1 = - 2 e x sin x = ( - 2 e x sin x ) - 2( e x cos x - e x sin x )+2( e x cos x ) = 0 y 0 2 = e x sin x + e x cos x y 00 2 = 2 e x cos x = (2 e x cos x ) - 2( e x sin x + e x cos x )+2( e x sin x ) = 0 Problem. 10. x 2 y 00 + xy 0 - y = lnx
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