The equation of the curve passing through origin and satisfying (1+x^2 ) dy d x +2 x y=4 x^2 is

Step-by-step solution

Rewriting the differential equation (1+x^2) dy dx + 2xy = 4x^2 in standard linear form yields: dy dx + 2x 1+x^2 y = 4x^2 1+x^2 . The integrating factor is e^ P(x)dx = e^ 2x 1+x^2 dx = e^ (1+x^2) = 1+x^2 . Multiplying through by the integrating factor: y(1+x^2) = 4x^2 dx = 4 3 x^3 + C . Given the curve passes through the origin, substituting x=0 , y=0 yields C=0 . The particular solution becomes y(1+x^2) = 4 3 x^3 , or equivalently 3y(1+x^2) = 4x^3 . This corresponds to option C .