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Mathematic Methods HW Solutions 45

Mathematic Methods HW Solutions 45 - Chapter 8 x 2 sin x x...

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Chapter 8 45 12.13 y = braceleftBigg x - 2 sin x, x<π/ 4 π 2 - x - 2 cos x, x>π/ 4 12.15 y = x sinh x - cosh x ln cosh x 12.16 y = - x ln x - x - x (ln x ) 2 / 2 12.17 y = - 1 4 sin 2 x 12.18 y = x 2 / 2 + x 4 / 6 13.1 y = - 1 3 x - 2 + Cx linear 1st order 13.2 (ln y ) 2 - (ln x ) 2 = C separable 13.3 y = A + Be - x sin( x + γ ) 3rd order linear 13.4 r = ( A + Bt ) e 3 t 2nd order linear, a = b 13.5 x 2 + y 2 - y sin 2 x = C exact 13.6 y = Ae - x sin( x + γ ) 2nd order linear, complex a,b,c +2 e x + 3 xe - x sin x 13.7 3 x 2 y 3 + 1 = Ax 3 Bernoulli, or integrating factor 1 /x 4 13.8 y = x ( A + B ln x ) + 1 2 x (ln x ) 2 Cauchy 13.9 y ( e 3 x + Ce - 2 x ) + 5 = 0 Bernoulli 13.10 u - ln u + ln v + v - 1 = C separable 13.11 y = 2 x ln x + Cx linear 1st order, or homogeneous 13.12 y = A ln x + B + x 2 y missing, or Cauchy 13.13 y = Ae - 2 x sin( x + γ ) + e 3 x 2nd order linear, complex a,b 13.14 y = Ae - 2 x sin( x + γ ) 2nd order linear, complex a,b,c + xe - 2 x sin x 13.15 y = ( A + Bx ) e 2 x + 3 x 2 e 2 x 2nd order linear, c = a = b 13.16 y = Ae 2 x + Be 3 x - xe 2 x 2nd order linear, c = a negationslash = b 13.17 y 2 + 4 xy - x 2 = C exact, or homogeneous 13.18 x = ( y + C ) e - sin y linear 1st order for x ( y ) 13.19 ( x + y ) sin 2 x = K separable with u = x + y 13.20 y = Ae x sin(2 x + γ ) 2nd order linear, complex a,b,c + x + 2 5 + e x (1 - x cos 2 x ) 13.21 x 2 + ln(1 - y 2 ) = C separable, or Bernoulli 13.22 y = ( A + Bx ) e 2 x + C sin(3 x + γ ) 4th order linear 13.23 r = sin θ [ C + ln(sec θ + tan θ )] 1st order linear 13.24 y 2 = ax 2 + b separable after substitution 13.25 x 3 y = 2 13.26 y = x 2 + x 13.27 y = 2 e 2 x - 1 13.28 y 2 + 4( x - 1) 2 = 9 13.29 62 min more 13.30
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