Question

Determine whether the given differential equation is exact. If it is exact, solve it.

i) (x 3 + y 3 )dx + 3xy2 dy = 0, Ans. x 4 + 4y 3x = C

ii) (y ln y − e −xy) + (1 y + x ln y) dy dx = 0, Ans. not exact

iii) (e −x sin y − 3)dx − (3x 2 − e x sin(2y))dy = 0, Ans. not exact

iv) (xy − 1)dx + (x 2 − xy)dy = 0, Ans. not exact.

4. Show that the equation (y ln y)dx + (x − ln y)dy = 0 is not exact. Find an integration factor to find a one parameter family of solutions. Ans: 2x ln y − (ln y) 2 = C.

Answer 1