常微习题讲解3.pdf

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1、z{!(/#a^100875)(2010~11_23)"(.}/$w

2、Æv/$,#

3、%~'•G.-7#=
120"Y

4、−1Rp(x)dxy=ze2,(4)0(1)℄z′′+I(x)z=0,(5)n121′I(x)=q(x)−p(x)−p(x)422yxc,(cI(x)8oG(1)I(x)=x22#);(2)I(x)=ϕ′′(x)−;(3)I(x)=const.,b!0(5)"Y

5、??)b!0(5)",ZbG(3)I(x)=const.,0(5)7#=
120b(5)"Y

6、8)e7#"=
120,Z2z=C1sinx+C2cosx.'0(7)",Z21y=√(C1sinx+C2cosx),xnC1,C22S#("G[Ww#">-+&-)(*%'3′′2′0.25UHrd
9y+xy+y=0.p−1Rp(x)dxzy=ze2=,x0z′′+z=0.Zz=C1cosx+C2sinx.''0",Z2cosxsinxy=C1+C2,xxnC1,C2S#•)hk3*℄`q8S-Lk3*[p(x)y′]′+[q(x)+λr(x)]y=0,(9)

7、R1(y)=R2(y)=03Y1oZnr(x)>0S[a,b]"iB#~&0.1w>λ=λ0x4(9)ey=φ0(x),7λ04*+,F7y=φ0(x)*.:y=φ0(x)>VZ)hkλ0")hB#)ZD#C6=0,y=Cφ0(x)>V")hB#8x8=Z)hkC)hB#
m"&f("iy`(>="[?,?,?].1~0.1wp(x)∈C([a,b]),q(x),r(x)∈C([a,b]),p(x)>0,r(x)>0,pα21+α226=0,β12+β22

8、6=0,(1)S-L4(9)qE(Y=)Fp_1imdtλ0<λ1<···<λk<···,limλk=∞;k→∞(2)λkS~φk(x)[a,b]lkeA(3){φk(x)}Dr(x)8[VZbr(x)φk(x)φm(x)dx=0,(k6=m);a4+&-)(*%'(4)2hg#Ri(ψ)=0(i=1,2)S~ψ∈C([a,b])C_0S~{φk(x)}8^D!}bW~X∞ψ(x)=akφk(x),k=0n"ZZ1bba=r(x)ψ(x)φ(x)dx,

9、2kδkFδk=r(x)φk(x)dx.kaa0.3

10、r4(y′′+λy=0,(10)y(0)=0,y(l)=0TS~(1)λ<0(10)n"120",Z2√√y=Ce−λx+Ce−−λx.12gWk+U!(C1+C2=0,√√C1e−λl+C2e−−λl=0.X.!C1=C2=0,(10)lYoZ'5"#λ(Æ6k3*")hk(2)λ=0(10)n"120",Z2y=C1x+C2.gWk+U!C1=C2=0.'λ=0Æek3*")hk(3)λ>0(10)n"120

11、",Z2√√y=C1cosλx+C2sinλx.gWk+U!(y(0)=C1=0,√√y(l)=C1cosλl+C2sinλl=0.+&-)(*%'5U2?1oZZ60#C1CC2Æ#ZoX.!√sinλl=0.√U4vYλl=nπ,Rnπ