Determine whether the given differential equation is exact. If it is exact, solve it. i) (x...

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.

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Colby Messinger answered 1 year ago

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