Verify that the given differential equation is not exact. (? xy sin x + 2 y cos

Question

Verify that the given differential equation is not exact.

(?xy sin x + 2y cos x) dx + 2x cos x dy = 0

If the given DE is written in the form

M(x, y) dx + N(x, y) dy = 0,

one has

Since

My

and

Nx

  ---Select--- are are not equal, the equation is not exact.

Multiply the given differential equation by the integrating factor

?(x, y) = xy

and verify that the new equation is exact.

If the new DE is written in the form

M(x, y) dx + N(x, y) dy = 0,

one has

Since

My

and

Nx

  ---Select--- are are not equal, the equation is exact.

Solve.

please explain im having trouble getting the right answers.

Explanation / Answer

(?xy sin x + 2y cos x) dx + 2x cos x dy = 0

My=(?x sin x + 2 cos x)

Nx=(2 cosx - 2 x sinx)

are are not equal, the equation is not exact

IF =xy

xy*[(?xy sin x + 2y cos x) dx + 2x cos x dy] =xy* 0

[-x^2y^2sinx+2xy^2cosxdx]dx +[2x^2 y cos x ]dy=0

M=[-x^2y^2sinx+2xy^2cosx]

My=[-2x^2ysinx+4xycosx]

N=[2x^2 y cos x ]

Nx=[4xycosx-2x^2ysinx ] =[-2x^2ysinx+4xycosx]

My=Nx ==>equation is exact

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