y/x=u
y=ux
dy/dx=u'x+u
åå¼=(y/x+æ ¹å·(1+(y/x)^2)=dy/dx
æ
ï¼u'x+u=u+æ ¹å·(1+u^2)
du/dx=æ ¹å·(1+U^2)
1/æ ¹å·(1+u^2) du=dxããããããããã1
ï¼å·¦è¾¹è®¾u=tant ,åï¼du=sec^2tdt å·¦=sectdt 积åå¾ï¼ln[(1+sint)/cost]=ln[sect+tant]
u^2=tan^2t=sec^2t-1 , sect=æ ¹å·(u^2+1)
æ
ï¼ln[sect+tant]=ln[u+æ ¹å·(u^2+1)]=ln[y/x+æ ¹å·[(y/x)^2+1])
æ
1å¼ä¸¤è¾¹ç§¯åå¾ï¼
ln[y/x+æ ¹å·[(y/x)^2+1]]=x+C
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