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

Step-by-step solution

Transform the differential equation into standard linear form: The given equation (1+x^2) dy dx + 2xy = 4x^2 divides by (1+x^2) to yield dy dx + 2x 1+x^2 y = 4x^2 1+x^2 . Compute the integrating factor: The integrating factor is e^ P(x) dx = e^ 2x 1+x^2 dx = e^ (1+x^2) = 1+x^2 . Multiply through and integrate: Multiply the standard form by the integrating factor: d dx [y(1+x^2)] = 4x^2 . Integration yields y(1+x^2) = 4 3 x^3 + C . Apply the initial condition: The curve passes through the origin, so y=0 at x=0 , giv