Differential form of a differential equationBy multiplying the differential equation6xydydx+x2+y2=0by dx we see that the equation has the alternative differential form(x2+y2)dx+6xydy=0.

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Step 1When dx and dy are differential forms the equation(x2+y2),&amp;nbsp;dx&amp;nbsp;+6xy,&amp;nbsp;dy&amp;nbsp;=0says that along every (not necessarily closed) curve in&amp;nbsp;R2&amp;nbsp;the integral of the vector field(x2+y26xy)must be zero. This means: if the curve isStep 2γ:t↦(γ1(t)γ2(t)) ,&amp;nbsp;&amp;nbsp;t∈[0,T]then&amp;nbsp;∫γ(x2+y2),&amp;nbsp;dx&amp;nbsp;+6xy,&amp;nbsp;dy&amp;nbsp;=∫0T(γ12(t)+γ22(t))γ˙1(t)+6γ1(t)γ2(t)γ˙2(t),&amp;nbsp;dt&amp;nbsp;must be zero. In particular, if the curve isγ:t↦(tγ2(t)) ,&amp;nbsp;&amp;nbsp;t∈[0,T]then&amp;nbsp;∫γ(x2+y2),dx+6xy,dy=∫0Tt2+γ22(t)+6,t,γ2(t),γ˙2(t),dt&amp;nbsp;(1)must be zero. Because this holds for every&amp;nbsp;T≥0&amp;nbsp;it follows that the integrand on the RHS must be zero. When we change notation this is nothing else than the ordinary differential equationx2+y2+6xydydx=0&amp;nbsp;(2) we started with.Conversely, using the same formalism it is easy to see that if y is a solution of the ODE (2) then the line integral (1) is zero along every curve. This is another way of saying that the differential form&amp;nbsp;(x2+y2),&amp;nbsp;dx&amp;nbsp;+6xy,&amp;nbsp;dy&amp;nbsp;&amp;nbsp;must be zero.

Differential form of a differential equation By multiplying the

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