SpecialIntegratingFactors

# SpecialIntegratingFactors - Special Integrating Factors...

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Special Integrating Factors Given the O.D.E. M(x,y) dx + N(x,y) dy = 0, assume it is non-exact. Suppose that n(x,y) is an integrating factor of the equation, then n(x,y) M(x,y) dx + n(x,y) N(x,y) dy = 0 is an exact equation. Therefore, ! ! y n(x,y)M(x,y) [ ] = ! ! x n(x,y)N(x,y) [ ] or n(x,y) ! M(x,y) ! y + ! n(x,y) ! y M(x,y) = n(x,y) ! N(x,y) ! x + ! n(x,y) ! x N(x,y) or n(x,y) ! M ! y " ! N ! x # \$ % ( = N ! n ! x " M ! n ! y (1) n(x,y) is an unknown function that satisfies equation (1), but equation (1) is a partial differential equation. So, in order to find n(x,y) we have to solve a P.D.E. and we do not know how to do it. Therefore, we have to impose some restriction on n(x,y). Assume that n is function of only one variable, let’s say of the variable x, then n(x) and ! n ! y = 0, ! n ! x = dn dx So, equation (1) reduces to n(x) ! M(x,y) ! y " ! N(x,y) ! x # \$ % ( = N(x,y) dn dx or 1 N(x,y) ! M(x,y) ! y " ! N(x,y) ! x # \$ % ( dx = dn n If the left hand side of the above equation is only function of x, then the equation is

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SpecialIntegratingFactors - Special Integrating Factors...

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