Re: [理工] [工數]ODE解不出來
※ 引述《bboycookie (閉關BBOY餅乾)》之銘言：
: q:
: cosy dx-2(x-y)siny dy -cosy dy=0
: ans:
: 2 2
: xcos y - ycos y =c
使用 Exact ODE Method
原式： cosy dx + [-2xsiny+2ysiny-cosy] dy = 0
M(x,y) = cosy
N(x,y) = -2xsiny+2ysiny-cosy
dM dN
--- = -siny ； --- = -2siny
dy dx
∵ dM/dy 不等於 dN/dx
所以還不是 Exact ODE
要再乘以一個integrator factor μ
μ(y) = exp[(dN/dx-dM/dy)/M] = cosy
μM(x,y) = cos^2y
又 df(x,y)/dx = μM(x,y)
↑
(應該是partial才對，因為partial的符號打不出來)
所以 f(x,y) = xcos^2y + g(y)
又df(x,y)/dy = -2xcosysiny + g'(y)
= μN(x,y)
= -2xcosysiny + 2ysinycosy + cos^2y
g'(y) = 2ysinycosy + cos^2y
g(y) = -0.5ycos2y - 0.5y = -0.5y(cos2y + 1) = -ycos^2y
f(x,y)= xcos^2y - ycos^2y = c
有錯請還煩請高手不吝指正 <(_ _)>
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