Re: [理工] [工數]-高階非線性O.D.E
※ 引述《zendla (夏夜薄荷)》之銘言：
: 出現三角的題目，感覺不是很好做啊，請大大們教一下
: Find the solution of the nonlinear ordinary differential equation
: y" + siny = 0
: with initial conditions: y(0) = 0, y'(0) = 2. [清大工科]
: y y
: ans:ln|sec── + tan──| = x
: 2 2
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感覺這題作法很多樣化
我是這樣想:
y" + siny = 0
→ y'(y'' + siny) = 0 for y'≠0
(y')^2
→ [ ______ - cosy ]' = 0
2
→ (y')^2 - 2cosy = c1
把 x=0 帶入:
2^2 - 2cos0 = c1 → c1 = 2
所以 (y')^2 = 2cosy + 2
→ y' = ± 2cos(y/2)
→ ∫ (1/2)sec(y/2) dy = ∫ ±1 dx
→ ln｜sec(y/2) + tan(y/2)｜ = ±x + c2
再由 y(0)=0 可解出 c2=0
y'(0)=2 可知 ln｜sec(y/2) + tan(y/2)｜ = -x 不合
因此 ln｜sec(y/2) + tan(y/2)｜ = x
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