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

Step-by-step solution

aligned & x y ~d y ~d x =x^2+2 y^2 & ~d y ~d x = x^2+2 y^2 x y ...(i) & Put y= v x...(ii) aligned Differentiating w.r.t. x , we get d y ~d x = v +x dv ~d x ...(iii) Substituting (ii) and (iii) in (i), we get array ll & v +x dv ~d x = x^2+2 v ^2 x^2 x( v x) & v +x dv ~d x = x^2 (1+2 v ^2 ) x^2 v & x dv ~d x = 1+2 v ^2 v - v = 1+ v ^2 v & v 1+ v ^2 dv = 1 x ~d x array Integrating on both sides, we get aligned & 1 2 2 v 1+ v ^2 dv = d x x & 1 2 |1+ v ^2 |= |x|+ | c ₁ | aligned aligned |1+ v ^2 | & =2 |x|+2 |c₁ | & = |