双曲函数及其在微分方程中的应用

4/2010双曲函数及其在微分方程中的应用(201620),,:;;,f(x)=f(x)+f(-x)2+f(x)-f(-x)2.F(X)=ex,ex=ex+e-x2+ex-e-x2,Ex,excoshx=ex+e-x2,sinhx=ex-e-x2,tanhx=sinhxcoshx=ex-e-xex+e-x;cothx=coshxsinhx=ex+e-xex-e-x;sechx=1coshx=2ex+e-x;cothx=1sinhx=2ex-e-x:(sinhx)=coshx;(coshx)=sinhx;(tanhx)=seeh2x;(cothx)=-csch2x.(JacobBernoull,i16541705):,,..,,1690.,(JohannBernoull,i1667-1748),.1[1],Ayy=f(x),,AM.:,,H;M,T,;,,p=s,s161,Tcos=H,Tsin=s,tan=1as,a=Htan=y,dydx=1as,s,x,d2ydx2=1adsdx,dsdx=1+dydx2,,d2ydx2=1a1+dydx2.OAa,y|x=0=a,y|x=0=0d2ydx2=1a1+dydx2y|x=0=a,y|x=0=0,yy=p,p(0)=0,d2ydx2=1a1+dydx2p=1a1+p2,ln(p+1+p2)=xa+C,p(0)=0C1=0,,p+1+p2=e-xa,1+p2-p=e-xa,,p=12exa-e-xa,y==12exa-e-xay|x=0=a,y=a2exa+e-xa.,m,,mg-kv2,(),F=ma,mdvdt=mg-kv2,[2]v=mgktanhgkmtlimtv=limtmgktanh(gkmt)mgkm=60,g=9.8217,,,[1].()[M].6.:2007.[2]MauriceWeir.ThomasCallculus(tenthedition)[M].,,,.:,2003.(12);,,,:;,;,,,[1].[M].:,2000.[2].[J].,1996(3).[3].[J].,1997(4).18