作者iyenn (晓风)
看板Grad-ProbAsk
标题Re: [理工] [工数]-一阶ODE
时间Wed Oct 21 13:37:51 2009
※ 引述《sususki (小雨)》之铭言：
: 请教各位工数大大..有个问题想不通
: 我是看周易的工数(p 1-73)
: 题目是：y'=y^2-xy+1
: 首先令通解 y=x+v ，代入ODE後 ==>1+v'=(x+v)^2-x(x+v)+1
: 左边的 1+v'是怎麽来的？？
y'=P(x)y^2+Q(x)y+R(x) ...(1)
if find the yp s.t yp'=y'=P(x)yp^2+Q(x)yp+R(x)
and we can let y=yp+v
y'=yp'+v'
yp'+v'=P(yp^2+v^2+2ypv)+Q(yp+v)+R
--->v'=Pv^2+(2Pyp+Q)v
--->v'-(2Pyp+Q)v=Pv^2 ....Bernoulli ode
let z=1/v z'=-v^-2v'
--->z'+(2Pyp+Q)z=-P .......1st linear ode
imply we can let y=yp+1/z ,change (1) to 1st linear ode
or y=yp+v ,change (1) to Bernoulli ode
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※ 编辑: iyenn 来自: 123.193.214.165 (10/21 13:42)