Re: [理工] 工程數學 ODE

[Honor1984 (希望願望成真)]

※ 引述《Spin9527 (9527)》之銘言：
: 我同事最近要考在職碩士專班
: 因為離學生時代有點久
: 所以請我代PO幾題工數與微積分的問題
: 煩請版上好心人幫忙詳解
: http://imgur.com/lGO1wyQ
: 感謝幫忙

(7)

dy         dx
---   =  -----
2y+1     3x+4


=> (1/2)ln(2y+1) = (1/3)ln(3x+4) + c

=> √[2y + 1] = C(3x+4)^(1/3)

(8)

u = -2x + y

u' = -2 + y'

u' + 2 = u^2 - 7

=> (1/6)ln[│(u-3)/(u+3)│] = x + c

=> │[-2x + y - 3]/[-2x + y + 3]│ = Aexp(6x)

y(0) = 0 => A = 1

[2x - y + 3] = exp(6x)[-2x + y + 3]

(9)

y = Acos(3x) + Bsin(3x)

代入y(0) = 0, y'(0) = 1

解得A = 0, B = 1/3

=> y = (1/3)sin(3x)

(10)

y = Aexp(3x) + Bxexp(3x)

代入y(0) = 1, y'(0) = 0

解得A = 1, B = -3

y = exp(3x) - 3xexp(3x)

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