常微分方程（吉林大学） CAD1

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1、5~©§6>f()¯Ìo]IKJJIIJI3ÆêÆÆ11
19£¶w«'4òÑ1ÙÐÈ©{¯ÌIKJJIIJI12
19£¶w«'4òÑ1ÙÐÈ©{§2;;;...§§§))){{{¯ÌIKJJIIJI12
19£¶w«'4òÑ1ÙÐÈ©{§2;;;...§§§))){{{2.45§¯ÌIKJJIIJI12
19£¶w«'4òÑ1ÙÐÈ©{§2;;;...§§§))){{{2.45§5§:dy¯Ì+p(x)y=q(x),(2.11)dxIKÙ¥¼êp(x)Úq(x)3«mIþëY.JJIIJI12

2、19£¶w«'4òÑ1ÙÐÈ©{§2;;;...§§§))){{{2.45§5§:dy¯Ì+p(x)y=q(x),(2.11)dxIKÙ¥¼êp(x)Úq(x)3«mIþëY.JJIIJIq(x)≡0,§12
19dy+p(x)y=0(2.12)£dx¶w«¡àg5§.'4òÑ1ÙÐÈ©{§2;;;...§§§))){{{2.45§5§:dy¯Ì+p(x)y=q(x),(2.11)dxIKÙ¥¼êp(x)Úq(x)3«mIþëY.JJIIJIq(x)≡0,§12
19dy+p(x)y=0(2.12)£dx¶w

3、«¡àg5§.'4q(x)6≡0,§(2.11)¡àg5§.òÑ©Û:¯ÌIKJJIIJI13
19£¶w«'4òÑ©Û:(2.12)´Cþ©l§,ÙÏ)R−p(x)dxy=Ce,Ù¥C?¿~ê.¯ÌIKJJIIJI13
19£¶w«'4òÑ©Û:(2.12)´Cþ©l§,ÙÏ)R−p(x)dxy=Ce,Ù¥C?¿~ê.¦)§(2.11),Cz=S(x)y,(2.13)¯ÌÙ¥S(x)´½¼ê.IKJJIIJI13
19£¶w«'4òÑ©Û:(2.12)´Cþ©l§,ÙÏ)R−p(x)dxy=Ce,Ù¥C?¿~ê.¦)§(2.11)

4、,Cz=S(x)y,(2.13)¯ÌÙ¥S(x)´½¼ê.IKò(2.13)(2.11),JJIIJIdzdy0=S(x)y+S(x)13
19dxdx=S0(x)y+S(x)(−p(x)y+q(x))£=(S0(x)−p(x)S(x))y+q(x)S(x).¶w«'4òÑ=dz0=(S(x)−p(x)S(x))y+q(x)S(x).dx¯ÌIKJJIIJI14
19£¶w«'4òÑ=dz0=(S(x)−p(x)S(x))y+q(x)S(x).dxXJUÀS(x),¦0S(x)−p(x)S(x)=0,(2.14)KÈ©§¯Ìd

5、z=q(x)S(x).IKdxJJIIJI14
19£¶w«'4òÑ=dz0=(S(x)−p(x)S(x))y+q(x)S(x).dxXJUÀS(x),¦0S(x)−p(x)S(x)=0,(2.14)KÈ©§¯Ìdz=q(x)S(x).IKdxJJII(2.14)JIdS14
19=p(x)S.dx£¶w«'4òÑ=dz0=(S(x)−p(x)S(x))y+q(x)S(x).dxXJUÀS(x),¦0S(x)−p(x)S(x)=0,(2.14)KÈ©§¯Ìdz=q(x)S(x).IKdxJJII(2.14)JIdS1

6、4
19=p(x)S.dx£¶w«§´'uSCþ©l§,)R'4p(x)dxS=Ce,òÑÙ¥C?¿~ê.){:¯ÌIKJJIIJI15
19£¶w«'4òÑ){:ÀRp(x)dxS=e.¯ÌIKJJIIJI15
19£¶w«'4òÑ){:ÀRp(x)dxS=e.C(2.13),K§(2.11)zdzRp(x)dx=q(x)e.dx¯ÌIKJJIIJI15
19£¶w«'4òÑ){:ÀRp(x)dxS=e.C(2.13),K§(2.11)zdzRp(x)dx=q(x)e.dxÈ©ZR¯Ìp(x)dxz=C+q(x)edx.I

7、KJJIIJI15
19£¶w«'4òÑ){:ÀRp(x)dxS=e.C(2.13),K§(2.11)zdzRp(x)dx=q(x)e.dxÈ©ZR¯Ìp(x)dxz=C+q(x)edx.IKJJII£Cþ¿n,=(2.11)Ï)LªZJIzRR−p(x)dxp(x)dx15
19y==eC+q(x)edx,(2.15)S(x)£Ù¥C?¿~ê.¶w«'4òÑeRxp(t)dtS=ex0,Ù¥x0∈I.¯ÌIKJJIIJI16
19£¶w«'4òÑeRxp(t)dtS=ex0,Ù¥x0∈I.K3C(2.13)e,§(2

8、.11)zRdzxp(t)dt=q(x)ex0.dx¯ÌIKJJIIJI16
19£¶w«'4òÑeRxp(t)dtS=ex0,Ù¥x0∈I.K3