6. `dy/(3x^2)=dx/(1-e^-y)`

Solution of the equation (dy)/(dx)=e^(x-y)(1-e^y) is

If e^(x)+e^(y)=e^(x+y) , prove that : (dy)/(dx)=-(e^(x)(e^(y)-1))/(e^(y)(e^(x)-1)) .

If e^(y)(x+1)=1, show that (dy)/(dx)=-e^(y)

Solve: (dy)/(dx)=e^(x-y)+x^2e^(-y)

If e^(x)+e^(y)=e^(x+y), prove that (dy)/(dx)=-(e^(x)(e^(y)-1))/(e^(y)(e^(x)-1)) or,(dy)/(dx)+e^(y-x)=0

Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as Solution of the differential equation (dy)/(dx)=e^(3x-2y)+x^2e^(-2y) is (e^(2y))/(2)=(e^(3x))/(3)+(x^2)/(2)+C Reason (R) : (dy)/(dx)=e^(3x-2y)+x^2e^(-2y) (dy)/(dx)=e^(-2y)(e^(3x)+x^2) separating the variables e^(2y)dy=(e^(3x)+x^2)dx int e^(2y)dy=int(e^(3x)+x^2)dx (e^(2y))/(2)=(e^(3x))/(3)+(x^3)/(3)+C .

The solution of the differential equation (dy)/(dx) = e^(3x-2y) +x^(2)e^(-2y) ,is

(dy)/(dx)=(x+e^(2x))/(y)

If e^y(x+1)=1 . Show that (d^2y)/(dx^2)=((dy)/(dx))^2