how the equation is not exact(5x^2y+6x^2y^3+4xy^2)dx+(2x^3+3x^4y+3x^2y)dy=0?
- 9 years agoFavorite Answer
Consider the equation (5x^2y+6x^3y^2+4xy^2)dx+(2x^3+3x^4y+3x^2y)dy=0 (a) show that the equation is not exact (b) Multiply the equation by x'' and y''' and determine values for n and m that make the resulting equation exact. (c) Use the solution of the resulting exact equation to solve the original equation.
- 7 years ago
∂M/∂y (5x^2y+6x^3y^2+4xy^2) =/= ∂N/∂x (2x^3+3x^4y+3x^2y
5x^2+12x^3y+8xy =/= 6x^2+12x^3y+6xy
Take x^ny^m and multiply it through the equation:
(5x^2y+6x^3y^2+4xy^2)dx+(2x^3+3x^4y+3x^2y)dy X x^ny^m
Now for example, yy^m is the same as y^(m+1) and y^2y^m is y^(m+2). So do this throughout the equation, after; we derive the partial derivative based on these values:
Now the coefficients need to be equal, since the x and y powers are equal:
5(m+1) + 6(m+2) + 4(m+2) = 2(n+3) + 3(n+4) + 3(n+3)
15m + 1 = 8n
m = 1, n =2 satisfies this!
Now plug it in to get an exact equation:
(5x^4y^2 + 6x^5y^3 + 4x^3y^3)dx + (2x^5y + 3x^6y^2 + 3x^4y^2)dy
Now solve it like any other exact equation. I'll give the solution I got (doesn't mean my solution is correct);
My Solution: x^5y^2+x^6y^3+x^4y^3+y
- Anonymous9 years ago
if ∂N/∂x =/= ∂M/∂y, then it's not exact
in other words, the partial derivatives of the each of the expressions (namely M & N) with respect to x, & y should be exactly the same, to be considered exact, then you can determine an arbitrary function Ψ.
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- exyLv 44 years ago
F*cking genuine. i truly love math, algebra, calculus, geometry, and so on. and that i might take a seat and try this each and every at some point if i had to. in fact I even locate new the thank you to choose new formula, and equations. i assume that is why I have been given all "A's" in college