Question

求xdy/dx+1=e^y通解

Answer 1

∵xdy/dx+1=e^y

==>xdy/dx=e^y-1

==>dy/(e^y-1)=dx/x

==>-e^(-y)dy/(e^(-y)-1)=dx/x

==>d(e^(-y)-1)/(e^(-y)-1)=dx/x

==>ln│e^(-y)-1│=ln│x│+ln│C│ (C是常数)

==>e^(-y)-1=Cx

==>e^(-y)=Cx+1

==>(Cx+1)e^y=1

∴原方程的通解是(Cx+1)e^y=1.