Question Number 114980 by srijachakraborty last updated on 22/Sep/20

Answered by Dwaipayan Shikari last updated on 22/Sep/20

(y^4 −2x^3 y)+(x^4 −2xy^3 )(dy/dx)=0  (dy/dx)=((2x^3 vx−v^4 x^4 )/(x^4 −2v^3 x^4 ))        (y=vx  v+x(dv/dx)=((2v−v^4 )/(1−2v^3 ))  (dv/dx)=((2v−v^4 )/(1−2v^3 ))−v  (dv/dx)=((v+v^4 )/(1−2v^3 ))  ∫(((1−2v^3 )dv)/(v+v^4 ))=∫dx  ∫((1+4v^3 )/(v+v^4 ))−((6v^2 )/((1+v^3 )))=x+C  log(v+v^4 )−2log(1+v^3 )=x+C  log(((v+v^4 )/((1+v^3 )^2 )))=x+C  ((v+v^4 )/((1+v^3 )^2 ))=Ce^x   (((y/x)+((y/x))^4 )/((x^3 +y^3 ))).x^6 =Ce^x   ((x^3 y+y^4 )/((x^3 +y^3 ))).x^2 =Ce^x   ((y+y^4 )/((1+y^3 )^2 ))=Ce  (y/(1+y^3 ))=Ce  (1/2)=Ce  C=(1/(2e))  ((x^5 y+x^2 y^4 )/((x^3 +y^3 )^2 ))=(1/2)e^(x−1)