Differential Equations Lecture Work Solutions 191

Differential Equations Lecture Work Solutions 191 - cos 2 x...

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1f. sin 2 xu xx +s in2 xu xy +cos 2 xu yy = x A =s in 2 xB =s in2 x =2s in x cos xC =co s 2 x 4 = 0 parabolic dy dx = 2sin x cos x 2sin 2 x =co t x y =lns in x + c ξ = y ln sin x = cot y =1 η = x =0 η y =1 u x = cot xu ξ + u η η x = cot xu ξ u y = u ξ + u η u xx =( cot xu ξ ) x = 1 sin 2 x u ξ +cot 2 xu ξξ u xy = cot x ( u ξ ) y = cot x ( u ξξ + u ξη ) u yy = u ξξ +2 u ξη + u ηη L. H. S. = u ξ +s in 2 x cos 2 x sin 2 x u ξξ +2s in x cos x ( cot x )( u ξξ + u ξη ) +cos 2 x ( u ξξ +2 u ξη + u ηη ) L. H. S. =co s 2 xu
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Unformatted text preview: cos 2 x u η η + u ξ = x ln sin x = y − ξ = η − ξ sin x = e η − ξ ⇒ cos 2 x = 1 − sin 2 x = 1 − e 2( η − ξ ) x = arcsin e η − ξ [1 − e 2( η − ξ ) ] u η η + u ξ = arcsin e η − ξ 191...
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