①求通解[xsin(y/x)-ycos(y/x)]dx+xcos(y/x)dy=0②解初值xy'-y=xtany/x,y(1)=π/2

Question

①求通解 [xsin(y/x)-ycos(y/x)]dx+xcos(y/x)dy=0 ②解初值 xy'-y=xtany/x,y(1)=π/2

Answer 1

1.令y/x=t,则方程化为(xsint-xtcost)dx+xtcostdx+x^2costdt=0
即dx/x=-cottdt
解得y=xarcsin(C/x)
2.令y/x=t,则方程化为xdt=tantdx
解得y=xarcsinCx
令y(1)=π/2得y=xarcsinx