The solution of x d y d x =y+x e^ y / x with y(1)=0 is

Step-by-step solution

Given differential equation is gathered x d y d x =y+x e^ y / x d y d x = y x +e^ y / x gathered It is a homogeneous differential equation. array llrl & Put y=v x d y d x =v+x d v d x & & v+x d v d x = v x x +e^ v x / x & v+x d v d x & =v+e^v & x d v d x & =e^v array e^ -v d v= 1 x d x On integrating both sides, we get aligned -e^ -v & = x+c -e^ -y / x & = x+c aligned Given, y(1)=0 array rlrl & & e^ -0 / 1 & = 1+c & & -1 & =0+c c=-1 & -e^ -y / x & = x-1] & & 1 & = x+e^ -y / x array