作业帮解微分方程,如图.
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/02 12:14:15
解微分方程,如图.
解微分方程,如图.
解微分方程,如图.
令x=m-2,t=n+3.则dx=dm,dt=dn
代入原方程,化简得 dm/dn=2[m/(m+n)]².(1)
再令m=ny,则dm/dn=ndy/dn+y
代入方程(1),化简得 ndy/dn=-y(1+y²)/(1+y)²
==>(1+y)²dy/[y(1+y²)]=-dn/n
==>[1/y+2/(1+y²)]dy=-dn/n
==>ln│y│+2arctany=-ln│n│+ln│C│ (C是非零常数)
==>y*e^(2arctany)=C/n
==>m*e^(2arctan(m/n))=C
==>(x+2)*e^(2arctan((x+2)/(t-3)))=C
显然,x=-2.即可取C=0
故原方程的通解是(x+2)*e^(2arctan((x+2)/(t-3)))=C (C是任意常数).
把x的弄到一边，T的弄到一边。再积分