Скачать или смотреть (xy^2-x^2)dx+(3x^2y^2+x^2y-2x^3+y^2)dy=0

#nonexactequation #reducibletoexact


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Here is a video of solving non-exact equation, by reducing the given equation exact form. Have a little patience and watch the video till end.

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Chinnaiah Kalpana🍁




Note:

If (1/M)[(partial derivative of N w.r.t. x) - (partial derivative of M w.r.t. y)] = g(y) [i.e., a function of y only] (or) k [real number] ,
then exp(∫g(y)dy) (or) exp(∫kdy) is an integrating factor of Mdx+Ndy=0.

exp(log g(y)) = g(y)
& exp(k logy) = exp[log(y^k)] = y^k
where k is constant.


Working rule to solve Mdx+Ndy=0:

1. General equation is Mdx+Ndy=0 ......(i) Observe (partial derivative of M w.r.t. y) ≠
(partial derivative of N w.r.t. x), then (i) is Non-Exact.

2. Find (1/M)[(partial derivative of N w.r.t x) - (partial derivative of M w.r.t y)] and observe it as a function of y alone = g(y) or a real constant k.

3. Then exp(∫g(y)dy) or exp(∫kdy) is an Integrating factor of (i).

4. Multiplying (i) with I.F. to transform it into an exact equation of (i), M1dx+N1dy=0 ...(ii)

5. Solve (ii) to get the general solution of (i).





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