试从dx/dy=1/y'导出：d^3x/dy^3=3（y'')^2﹣y'y'''/(y')5

Question

试从dx/dy=1/y'导出：d^3x/dy^3=3（y'')^2﹣y'y'''/(y')5

Answer 1

dx/dy=1/y′
d²x/dy²=（d/dy）{dx/dy}=(d/dy)(1/y′)
={(d/dx)(1/y′)}dx/dy
={-y″/(y′)²}(1/y′)
=-y″/(y′)³
d³x/dy³=d/dy{d²x/dy²}=d/dx{d²x/dy²}dx/dy
=d/dx{-y″/(y′)³}(1/y′)
={-y′″(y′)³+y″3(y′)²y″}/ (y′)^7
=3（y'')^2﹣y'y'''/(y')5