Question

) Determine a function N(x,y) so that the following DE is exact? [y^1/2.x^-1/2+x/(x^2+y)]dx = −N(x, y)dy

Answer 1

`N(x,y) = sqrt(x/y)+ 1/(2(x^2+y))`

Explanation:

`f_x = y^(1/2)x^(-1/2)+x/(x^2+y)`

`implies f = 2 y^(1/2)x^(1/2)+1/2 ln (x^2+y) + alpha(y)`

`implies f_y = N(x,y) = y^(-1/2)x^(1/2)+ 1/(2(x^2+y)) + alpha'(y)`

Comparing the mixed partials:

• `f_(xy) = 1/2 y^(-1/2)x^(-1/2)- x/(x^2+y)^2`

• `f_(yx) = 1/2 y^(-1/2)x^(-1/2)- x/(x^2+y)^2`