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Determine whether the given differential equation is exact. If it is exact, solve it. i) (x 3...

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Advance Math

Determine whether the given differential equation is exact. Ifit 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. notexact

iii) (e ?x sin y ? 3)dx ? (3x 2 ? e x sin(2y))dy = 0, Ans. notexact

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 notexact. Find an integration factor to find a one parameter family ofsolutions. Ans: 2x ln y ? (ln y) 2 = C.

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