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yy-b-x-a-ay-2-1-x-2-




Question Number 128106 by bemath last updated on 04/Jan/21
 yy′ + b(x−a)= ((ay^2 )/(1+x^2 ))
yy+b(xa)=ay21+x2
Commented by mr W last updated on 05/Jan/21
once i asked this question due to  Q127997, but then i could solve it myself  and therefore have deleted the post.  my post existed only for a short time.  thank you that you have noticed my   post nevertheless and restored the  question here!  i enjoyed that i could solve  Q127997 completely. please check out  the solution there, comments are  welcome!
onceiaskedthisquestionduetoQ127997,butthenicouldsolveitmyselfandthereforehavedeletedthepost.mypostexistedonlyforashorttime.thankyouthatyouhavenoticedmypostneverthelessandrestoredthequestionhere!ienjoyedthaticouldsolveQ127997completely.pleasecheckoutthesolutionthere,commentsarewelcome!
Answered by liberty last updated on 04/Jan/21
 yy′+b(x−a)=((ay^2 )/(1+x^2 ))   yy′−((ay^2 )/(1+x^2 )) = b(a−x)   let y^2  = v ⇒2yy′ = v′ ; yy′ = ((v′)/2)  ⇒ ((v′)/2)−(a/(1+x^2 )).v = b(a−x) ; v′−((2a)/(1+x^2 )).v=2b(a−x)  put IF μ = e^(−∫ ((2a)/(1+x^2 )) dx) = e^(−2a .arctan x)   we get v = ((∫2b(a−x)e^(−2a.arctan x)  dx+C)/e^(−2.arctan x) )   ∴ y^2 = C.e^(2a.arctan x)  + 2b∫ (a−x).e^(−2a.arctan x)  dx
yy+b(xa)=ay21+x2yyay21+x2=b(ax)lety2=v2yy=v;yy=v2v2a1+x2.v=b(ax);v2a1+x2.v=2b(ax)putIFμ=e2a1+x2dx=e2a.arctanxwegetv=2b(ax)e2a.arctanxdx+Ce2.arctanxy2=C.e2a.arctanx+2b(ax).e2a.arctanxdx
Commented by mr W last updated on 05/Jan/21
for complete solution see Q127997.
forcompletesolutionseeQ127997.

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