If dy dx + ^2 x y ^2 y x = 0 , where x, y (0, 2 ) and y ( 4 ) = 4 , then the value of y ( 3 ) is equal to

Step-by-step solution

Given differential equation is: dy dx = - ^2 x y ^2 y x Separating the variables, we get: ^2 y y dy = - ^2 x x dx Integrating both sides: ^2 y y dy = - ^2 x x dx ( y) = - ( x) + C Given y ( 4 ) = 4 , we substitute x = 4 and y = 4 : ( 4 ) = - ( 4 ) + C (1) = - (1) + C C = 0 So, the equation of the curve is: ( y) + ( x) = 0 ( x y) = 0 x y = 1 To find y ( 3 ) , substitute x = 3 : ( 3 ) y = 1 3 y = 1 y = 1 3 Since y (0, 2 ) , we get y = 6 . Answer: 6