Question

do it fast with explaination

Answer 1

There are two methods to solve given question.
1. By doing variable separation dy/dx=y² so dy/y²=dx now integrating wee get required solution.
2.The easiest method to solve this is using the given condition of y(1)=1 and substituting x=1 and y=1 in given options and only option A satisfies the required condition so solution is 2y-xy=1.

Answer 2

dy/dx = y²

dy/y² = dx

y-²dy = dx

integrater both sides

y^(-2+1)/(-2+1) = x + C

y-¹ (-1) = x + C

1/y + x + C = 0

a/c to question,

y( 1) = 1

so,

1/y(1) + 1 + C = 0

C = -2

so,

solution of the differrentail equation , is
1/y + x - 2 = 0

1 + xy - 2y = 0

2y - xy = 1

hence, option ( a) is correct