The solution of d y d x + 1 x = e^y x^2 is

Step-by-step solution

Given differential equation, d y d x + 1 x = e^y x^2 Dividing the eqn. by e^y , we get Let e^ -y =v e^ -y d y d x = d v d x From eqn. (i), we get d v d x - 1 x v= -1 x^2 Integrating Factor, IF =e^ - 1 x d x =e^ - x =e^ 1 x = 1 x Now, the required solution is aligned & v( IF )= - 1 x^2 ( IF ) d x+C₁ & v ( 1 x )= -1 x^2 1 x d x+C₁ & v x =- x⁻³ d x+C₁ & v x = 1 2 x^2 +C₁ v x = 1+2 x^2 C₁ 2 x^2 & 2 x v=1+C x^2 [ 2 C₁=C ] & 2 x e^ -y =1+C x^2 [ v=e^ -y ] & 2 x= (1+C x^2 ) e^y [ ^y . aligned