Integrating factor
Example 1

For example, the differential equation

can be solved exactly with integrating factors. The appropriate p (x) {\displaystyle p(x)} can be deduced by examining the y ′ {\displaystyle y'} term. In this case, 2 p (x) = 2 x {\displaystyle 2p(x)=2x}, so p (x) = x {\displaystyle p(x)=x}. After examining the y {\displaystyle y} term, we see that we do in fact have p (x) 2 + p ′ (x) = x 2 + 1 {\displaystyle p(x)^{2}+p'(x)=x^{2}+1}, so we will multiply all terms by the integrating factor e ∫ x d x = e x 2 / 2 {\displaystyle e^{\int x\,dx}=e^{x^{2}/2}}. This gives us

which can be rearranged to give

Integrating twice yields

Dividing by the integrating factor gives:
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