1
y'+ytanx=sin2x
dy+ytanxdx=sin2xdx
cosxdy+ysinxdx=sin2xcosxdx
cosxdy+ydcosx=2cosx^2sinxdx
dycosx=(-2/3)dcosx^3
ycosx=(-2/3)(cosx)^3+C
2
yy''+1=y'^2
y'=p
y''=dp/dx=dp/dy*(dy/dx)=pdp/dy
ypdp/dy+1=p^2
pdp/(p^2-1)=dy/y
ln|p^2-1|=2ln|y|+lnC
p^2-1=Cy^2
p=√(1+Cy^2) 或 p=-√(1+Cy^2)
dy/√1+Cy^2=dx dy/√(1+Cy^2)=-dx
通解 通解x=(-1/√C)ln|y√C+√(1+Cy^2)|+C1
x=(1/√C)ln|y√C+√(1+Cy^2)|+C1
∫dy/√(1+Cy^2)
y=tanu/√C
=(1/√C)∫secudu
=(1/√C)ln|secu+tanu|=(1/√C)ln|y√C+√(1+Cy^2)|