高等数学微分方程问题

Question

xy'+2y=xlnx满足y(1)=-1/9的解为多少？最好有具体过程

Answer 1

解：设x=e^t,则dt/dx=1/x
∵y'=dy/dx=(dy/dt)(dt/dx)=(dy/dt)/x
==>xy'=dy/dt
∴代入原方程得dy/dt+2y=te^t
∵齐次方程dy/dt+2y=0的特征方程是r+2=0 ==>r=-2
∴齐次方程dy/dt+2y=0的通解是y=Ce^(-2t) (C是积分常数)
设y=(At+B)e^t是dy/dt+2y=te^t一个特解
∵y'=(At+B)e^t+Ae^t
代入dy/dt+2y=te^t
得(At+B)e^t+Ae^t+2(At+B)e^t=te^t
==>3Ate^t+(A+3B)e^t=te^t
==>3A=1,A+3B=0
==>A=1/3,B=-1/9
∴y=(t/3-1/9)e^t
∴dy/dt+2y=te^t的通解是y=Ce^(-2t)+(t/3-1/9)e^t
∴原方程的通解是y=C/x²+(lnx/3-1/9)x
∵y(1)=-1/9
∴C-1/9=-1/9 ==>C=0
故所求解是y=xlnx/3-x/9