공업수학공식정리

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공업수학공식총망라

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본문내용

idea y=eλx & λ2+aλ+b=0
실근
중근 (λ=-a/2)
허근
(λ=α±ωi )
중근 y2=uy1
y2′=u＇y1+uy1′
y2″=u＂y1+2u＇y1′+uy1″
(u＂y1+2u＇y1′+uy1″)+a(u＇y1+uy1′)+buy1=0
u＂y1+u＇(2y1′+ay1)+u(y1″+ay1′+by1)=0
u＂y1=0
u＂=0
u=c1x+c2
y=(c1+c2x)eλx
허근 y=c1eα+iω+c2eα-iω
eiωx=cosωx+isinωx
y=eαx[(c1+c2)cosωx+i(c1-c2)sinωx]
=eαx(Acosx+Bsinx)

x2y＂+ axy＇+ by = 0 동차미방

idea y=xm m2+(a-1)m+b=0
실근
중근 (m=(1-a)/2)
허근
(m=μ±νi )
중근 y2=uy1
y2′=u＇y1+uy1′
y2″=u＂y1+2u＇y1′+uy1″
x2(u＂y1+2u＇y1′+uy1″)+ax(u＇y1+uy1′)+buy1=0
u＂x2y1+u＇x(2xy1′+ay1)+u(x2y1″+axy1′+by1)=0
u＂x2y1+u＇xy1=0
, ln|u＇|=-lnx,
u=lnx
y=(c1+c2lnx)xm
허근 y=c1xμ+iν+c2xμ-iν
xiνx=xiνlne=eiνlnx=cos(νlnx)+isin(νlnx)
y=xμ[(c1+c2)cos(νlnx)+i(c1-c2)sin(νlnx)]
=xμ[Acos(νlnx)+Bsin(νlnx)]

참고 자료

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