Doubt regarding a differential equation.Solve: x,dx+y,dy=a2(x,dy−y,dx)x2+y2The author further proceeds to rearrange above in the form M,dx+N,dy=0 whereM=x+a2yx2+y2;N=y−a2xx2+y2Which further impliesMy=Nx=a2(x2−y2)(x2+y2)2This is the required condition for the given equation to be exact and the solution is obtained using standard formula.But, what I did is as follows:x,dx+y,dy=a2(x,dy−y,dx)x2+y2→(x2+y2)x,dx+(x2+y2)y,dy=a2(x,dy−y,dx)Therefore (x3+xy2+a2y),dx+(y3+yx2−a2x),dy=0Comparing it with the equation M,dx+N,dy=0 we get,M=x3+xy2+a2y;N=y3+yx2−a2xBut My=2xy+a2;Nx=2xy−a2

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Cody Hart · Accepted Answer

Step 1x,dx+y,dy=a2(x,dy−y,dx)x2+y2tag1As you correctly found : Equation (1) is an exact ODE.You transformed Eq.(1) into Eq.(2):(x3+xy2+a2y),dx+(y3+yx2−a2x),dy=0tag2Again you correctly found : Equation (2) is not an exact ODE.Step 2This is not contradictory because Eqs.(1) and (2) are not the same ODE.In order to transform Eq.(2) into an exact ODE one have to multiply it with an &quot;integration factor&quot;, say μ(x,y).One find that μ(x,y)=1x2+y2.μ(x,y)Big((x3+xy2+a2y),dx+(y3+yx2−a2x),dyBig)=0tag3x3+xy2+a2yx2+y2,dx+y3+yx2−a2xx2+y2,dy=0is exact. is exact.In fact Eq.(3) is equivalent to Eq.(1) thanks to the correct μ(x,y).

Aarlenlsi1 · Accepted Answer

Step 1In my mind, your question is about integrating factor.You have checked your equation(x3+xy2+a2y)dx+(y3+yx2−a2x)dy=0 is not exact.That&#039;s why we need integrating factor. Here the integrating factor is exactly 1x2+y2.Step 2By the integrating factor, we make your equation an exact equation deliberately, then we can solve the ODE by exact equations.If x2+y2=0, then this ODE is not well-defined.p.s. It&#039;s my first time to answer a problem. Perhaps I misunderstand your problem, please tell me and I would try to correct my answer.

Doubt regarding a differential equation. Solve: \(\displaystyle{x},{\left.{d}{x}\right.}+{y},{\left.{d}{y}\right.}={\frac{{{a}^{{2}}{\left({x},{\left.{d}{y}\right.}-{y},{\left.{d}{x}\right.}\right)}}}{{{x}^{{2}}+{y}^{{2}}}}}\) The author

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