Let x = x(y) be the solution curve to the differential equation y dx dy - 2x = x^2 + y^4 for y > 0 , satisfying the initial condition x(1) = 0 . Then the value of x(2) is equal to

Step-by-step solution

Given the differential equation: y dx dy - 2x = x^2 + y^4 Divide the entire equation by y^3 : y dx dy - 2x y^3 = 1 y^3 x^2 + y^4 y dx dy - 2x y^3 = 1 y ( x y^2 )^2 + 1 Let u = x y^2 . Differentiating with respect to y gives: du dy = y^2 dx dy - x(2y) y^4 = y dx dy - 2x y^3 Substituting this into the differential equation, we get: du dy = 1 y u^2 + 1 Separating the variables: du u^2 + 1 = dy y Integrating both sides: du u^2 + 1 = dy y |u + u^2 + 1 | = |y| + C u + u^2 + 1 = Cy Substitute u = x y^2 back: x y^2 + x^2 y