The term M(x,y)dx+N(x,y)dy becomes an exact differential if the left hand side above is divided by y^7. Integrating that new equation, the solution of the differential equation is

Question

The differential equation

y−4y^7=(y^6+6x)y′

can be written in differential form:

M(x,y)dx+N(x,y)dy=0

where

M(x,y)=

, and N(x,y)=

.

The term M(x,y)dx+N(x,y)dy becomes an exact differential if the left hand side above is divided by y^7. Integrating that new equation, the solution of the differential equation is

=C.

Answer 1

y−4y^7=(y^6+6x)y′; (y - 4y⁷)dx +(- y⁶ - 6x)dy = 0; (y^{- 6} - 4)dx + (- 1/y - 6x/y⁷)dy;

M(x, y) = y^{- 6} - 4; My= - 6y^{- 7;}

N(c x, y) = (- 1/y - 6xy^{- 7}. N_x = - 6y^{- 7}; So, we have now exact equation. ∂F/∂x = y^{- 6} - 4; F(x,y) = ∫(y^{- 6} - 4)dx =

y^{- 6}x - 4x + f(y); ∂F/∂y = - 6y^{- 7}x + f'(y) = - 1/y - 6xy^{- 7}; f'(y) = -1/y; f(y) = - lny + C

F(x, y) = x/y⁶ - 4x - lny + C