微分方程的例題分析及解法

微分方程的例題分析及解法yepxdxq(x)epxCy()yfxy令uuxx()yfx(,)fxyyp，y(,)yfyyypyp20pqf(x)eP(x)xm~(x)eP(x)cosxp(x)sinx和flnyf(x)g(y)f(x)g(y)0的微分方程稱為變量可分離的微分方程，或稱可分離變量的微分方程，若1122f(x)g(y)021g(y)f(x)21g(y)1f(x)2G(y)F(x)Cxc)y1yy()dyyyff()或xdxxyfu)ux()()ypxyqx(x)q(x)yy、，p(q(xep(x)C)yep(x)()0ypxyyC(x)epxdxCe()()ypxCC(x)C(x)yC(x)ep(x)dx。yxylnxdx(xlny)dy0x(y)xx1yyy()()ypxyqxy，n(0nyzn)p(x)zn)q(x)1zC(x)n，CC(x)C(x)于yp(x)y0yCep(x)dx(x)yx)ep(x)dxyp(x)yq(x)用CCC(x)C(x)()Cxep(x)dxqx()C(x)C(x)ep(x)dxC(x)C(x)解yC(x)。p(x)C(x)q(x)eDy21nx例求yxx12lnxyp(x)q(x)y0的解xxxy。yC(x)x2lnx()Cxxx2lnx12lnxd()C(x)x2x2222lnxdxlnxCxx2xx22(lnxC)x2xlnx2Cxyxx表yypf(x,p)ypyypdpypdyypf(y,p)()pyqyfx,pqy當f(x)0pq0,212f(x)0yyyy0ypyy(x)yfyy。表的形式q0(),()fx()同次的多項式f(x)n0yx(x)q0是特征方程的單根（即時pq0yx2(x)yQ(x)eammf(x)mm與多項式乘積yx2Q(x)emyA(x)cosxB(x)sinxllf(x)P(x)xlnQ(x)sinxyxA(x)cosxBxmllA(x、B(x)都是次多項式lllxBx()cos()sinxa不是特征根a是特征根f(x)e[p(x)xyaxnllQ(x)sinx]()cosxB(x)sinxeAxxyaxmll(x)ayfkkx2(x)Q(x)A(x)，y或或mlB(x)ylxCe1)1是CmCxCxemx12mai(cosecxCxsin)12y(4)2y2y2y6y022210432((0_22i123,41()CCxeix123,4cosx_Csinx,所以通解中含加項C34yCeCxeCcosxCsinxxx1234例1）(y)yyx(y)2ycosx222yy53ln2xxy2yxyyxx,yx；,xyC2；yy22yyxCxxy,yy,xy1；22；yy）dxdxxesintxtt；；xtdtdtdxdxxttxt2；2；2dtdt2cosxy（）yxx；yyexeCxCx)；x12(CcosxCsinx)xe(CcosxCsinx)；xex；2x12120yy）CeCey(CCx)e；yyxxx1212CxCxy(CCx)e；；x1212解(y)2和yy中(y)2D513lnxyx。yyxxBdx為t(xAdt(x)ecosxyyxe(CcosxCsinx),k由fxkx12aiaaie(CcosxCsinx),yB。x120yy的iyCxCxC正12例2。xye。x0yy(0)(0)1yy。y1y。x解xyx12121212yCy1Cyx2。x2yeyeCyeCxC,xxx1120yy20012為yCeCx12y(0)y(0)1C2ye2。Cx12y,xdu,,uln（4）設(shè)u則xxlny例3(ee)dx(ee)dy0；xyxyxyy2(x2y)dxxdy0；(yxy)dxxdyy0224x2y,y(0)1yxy2解e(edxe(edy0xyyxeexye1e1xyd(ed(eyxe1e1yxln(eln(eCyx1(eeCxydx、dy0x2當xyx2y)dxdy0xyy令uxdyudxxduuu)xdu)0u0du2dxu2x12lnxCu1yx2lnxCuxy1xy2lnCx(xy2dx、dy20當xxyy[1()20xxyxy令uu1u)dxudxxdu)02x1u2lnu1uln2u1u2y將uxyx10C,C1。yCx222y0yxyx222y(xy)x22221y(x22x(4y）2yx)2d(4y4y22)dx1x2)24y1x22ln(4y)x)lnC22(4yx)C22（Cy1C5x0(4yx)522例41xyxy(2xyxdx0y0，x2dy(23)1x3yx2yyy(xy)0yee0xyx解yyxxyep(x)[q(x)ep(x)C]p(x)x,q(x)x,ye[C]elncosx[secxelncosxdxC]cos[secxsecxdxC]xxC)sinxCcosx（Ctan0yyxy0將yy,xylnxC1CxCy11C(x)x,設(shè)y()cossec,()sec2CxxCxxx2C(x)secxdxtanxCy(tanxC)cosxsinxCcosx（Cdy211y2dxxxx211(x),q(x)pxxx21122ye[()]eCxxxx211e2lnx[()edxC]2lnxxx2111[()xdxC]2x2xx211[(xx)C]2x2211C2xxC21y0C2111y2x2x223x2y0yx3yCe23x2Ce1Cxe1dx3lnx3x3x2x21y1Ce11yx3exxyx(y)yy11x,(yy11p(y),q(y),yyyxe1[e11dydyC]dyylnyylnyy1elnlny[eC]lnlnyy1y[C]yy11(yC)2y21Cy（C2y(yee0xydy1eyedx,exx1eeyyln(eeCe1Cex，eyxyyx)（Ce()()yfxgyy()yfx()yfcuc[q(x)ep(x)C]yep(x)①()()ypxyqx②p(xq(x)yCep(x)dxC()CxyC(x)ep(x)dx③C(x)C(x)Cep(x)ep(x)q(x)ep(x)yC0例5yxe()0y2yyxx)y2xy,y(0)0,y(0)3y(y)y23,yyyy解yxedxxeeCxxx1y(xeeC)dxxx1xe2eCxCxx12pxyypp020pppyCyyCyp11yCxlnC12yCeCx120ypyCyCpyCecx1201yCeCx12,ypy設(shè)yp2xp1x2lnpx)lnC21pyC)x21y(0)3C31)yyx2x3xC32(0)1得C1y，1yx3x13yy與pyypxppp30當ppp12PyC,PyC)y將pyC)lnyC)xlnC即2sin(yC)Cex2yarcsin(Ce)C(CC)x211ypyppp3pp(p211)xC2p21即故p2Ce2x11edxxydxCe2x1Ce2x1arcsin(Ce)CCx121C()yfxypyxyyypxyypxy例640560yyyyyy解022iye2(Ccos3xCsin3x)x126026，111yCe6Cexx12例74y4yy0(0)y(0)0y解44102121,21y(CCx)ex21212y(0)y(0)0CC1211yx)ex22例82y5y5x2y6y9y(xe23xyyyyy2y5y020解y2為52125yCCex21205xea20xyx(Ax2BxC)e0Ax3Bx2Cxx(x)yQ(x)f2y3Ax2BxC2y6Ax2B15AxA10B)x(4BC)5x2215A512A10B04BC0128,B,CA3525128yxxx323525x(1285yCCexx2x)323512690yyy69023()yCCxe3特征根為兩個相符的實根x12a3yx2(Ax2BxC)e3x[3Ax(4a3B)x(3BC)xCxey4323xy[9Ax4(24A9B)x2A18B9c)x2(6B12C)xCe3x126Cx122112A,BC1211yx(x)e3x2212211yCCx)e3xx(x)e3x221221211(xxCxC)e423x12221例9816yxe4yyx22y,(0)(0)0yyyxeyx(x)xef(x)xf(x)e與4解f4xx12y8()yfxxyy①②1816()yfxx4yyx2yyy與yy1122020yAxB1y、y228A16Ax16Bx，11,BA163211y1x16324yCx2e42xy、y2212Cee,C4x4x1yx2e4x22111yy1y2xx24xe163221a2202y(AxB)exy(ABAx)exy(B2AAx)ex(AxB)exexxAB0yxexye(CcosxCsinx)x12yyye(CcosxCsinx)xexx12y(0)y(0)0CC012y(cosxx)ex例44ye2sin5xyyyx23(sinyxyx224cosxyyyex440解2212(x)esin5x,aai2if2xye2(Acos5xBsin5x)xsin5xf(x)Ae2xf(x)e2(sin5x0cos5x)xye2[(2A5B)cos5x(2B5A)sin5x]xye2[(20B21A)cos5x(20A21B)sin5x]x,yy將ye2(25Acos5x25Bsin5x)e2sin5xxx1B,A251ye2xsin5x25230,1f(x)(xsinx2屬12(x)e[p(x)cosxQ(x)sinx]ap(x)0于flny(axa)cosx(bxb)sinx1010,y,有yy(abbx)cosx(baax)sinx101101y(2abbx)sinx(2baax)cosx101101(4ab2ab4axbx)cosx001111(2a4b2a2b2axbx)sinx(xsinx0111114ab2ab000114ab0112ab2ab100112ab111115a,b1011111a,b25500011111y(x)cosx(x)sinx102555022021ifxex()4cosx（3）特征方程為~f(x)e型，這里ap(x)p(x)0,ai1nyxe(CcosxCsinx)x12(x)CxCx,為求導計算方便，設(shè)u12yxeuxye(uxuxu)xye(xu222)xuuxuux,uuxCxuC12y2y2ye(xuxu2)uxeux2e(CsinxCcosx)x124xxCC12y2xesinxxcos例yyy2x(x)sinx2解將f11f(x)cos2x22y0yyy①y121yyy②③cos2yyyx2yyyy和yy112213102i22331yeCx(xCx)222120yA11212yA1aiayBcos2xBsin2x212y、y221(3B2B)cos2x(3B2B)sin2xcos2x21221123B2B123B2B02131,B,B故26131231ycos2xsin2x26132yyyy1233131xeCxCx)2x2x222212例a2解設(shè)p(x,y)yf(x)Yyy(Xx)或Yy(Xx)y(X,Y)xX11Sx[y(Xx)ydX[y(Xx)yX]yxyx2x02220Sa2得1yxyxa2222y6a2y即xx22yCeCe2lnCx2xxyC(x)x26a26a2()CxxCx,()2x2x46a22a2C(x)Cx4x32a2y(C)x2x32a2y()2xx[x,xdx]dxydx例潛水艇下降過程中受到重力mgkdtkmd2x即2d2mkm①20km2k,mkxCCet②m12(x)mg,0fkAxAtmgmg,Akmgkxt③④kkxt)CCettm120x(0)x(0)0tx(0)CC012kkkx(0)(Cet)0mm2t0mg2m2gC,C2k21k2xtkmg2kxt)tet)mk2yx。