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Integrating Factor

Oleh
An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable. For example, a linear first-order ordinary differential equation of type
(dy)/(dx)+p(x)y(x)=q(x),
(1)
where p and q are given continuous functions, can be made integrable by letting v(x) be a function such that
v(x)=intp(x)dx
(2)
and
(dv(x))/(dx)=p(x).
(3)
Then e^(v(x)) would be the integrating factor such that multiplying by y(x) gives the expression
d/(dx)[e^(v(x))y(x)]=e^(v(x))[(dy(x))/(dx)+p(x)y(x)]
(4)
=e^(v(x))q(x)
(5)
using the product rule. Integrating both sides with respect to x then gives the solution 
y(x)=e^(-v(x))inte^(v(x))q(x)dx.  


Reference : http://mathworld.wolfram.com/IntegratingFactor.html







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