If the solution of d y d x = y^3 x x e^ 1 / y^2 , y(0)=1 is 1 y^2 = _e(f(x)) , then f(x)=

Step-by-step solution

d y d x = y^3 x x e^ 1 / y^2 e^ 1 / y^2 y^3 d y= x x d x On putting 1 y^2 =t - 2 y^3 d y=d t and on putting u= x d u= d x 2 x we get e^t d t (-2) = 2 u d u aligned & - 1 2 (e^t )=2 u+k & - 1 2 (e^ 1 / y^2 )=2 x +k aligned y(0)=1 means at x=0, y=1 - 1 2 (e^ 1 / y^1 )=2 0 +k k=- e 2 - 1 2 (e^ 1 / y^2 )=2 x - e 2 Takin, both sides with base e , ( e^ 1 / y^2 2 )= [(2 x )- e 2 ] aligned & (- 1 2 )+ (e^ 1 / y^2 )= (2 x - e 2 ) & 1- (-2)+ 1 y^2 = (2 x - e 2 ) & 1 y^2 = [ (2 x - e 2 ) (-2) ] & 1 y^2 = (e-4 x ) aligned f(x)