The solution of the differential equation (x^2+y^2 ) d x-5 x y ~d y=0, y(1)=0 , is :

Step-by-step solution

aligned & (x^2+y^2 ) d x=5 x y d y & d y d x = x^2+y^2 5 x y aligned Put y=V x aligned & V + x dv dx = 1+ V ^2 5 ~V & xdv dx = 1-4 ~V ^2 5 ~V & V 1-4 ~V ^2 dV = dx 5 x aligned Let 1-4 ~V ^2= t -8 ~V dV = dt aligned & dt (-8)( t ) = dx 5 x & -1 8 | t |= 1 5 | x |+ C & -5 | t |=8 | x |+ K & x ^8+ | t ^5 |+ K =0 & x ^8 | t ^5 |= C & x ^8 |1-4 ~V ^2 |^5= C & x ^8 | x ^2-4 y ^2 x ^2 |^5= C & | x ^2-4 y ^2 |^5= Cx x ^2 & given y (1)=0 & |1|^5= C C =1 & | x ^2-4 y ^2 |^5= x ^2 aligned