Let a curve y = f(x) satisfy the differential equation x dy dx - 2y = x^3 x for x > 0 . If the curve passes through the point ( , 0) , then the value of the definite integral ₀^ f(x) dx is equal to

Step-by-step solution

The given differential equation is: x dy dx - 2y = x^3 x Dividing by x , we get a linear differential equation: dy dx - 2 x y = x^2 x The integrating factor (I.F.) is: I.F. = e^ - 2 x dx = e^ -2 x = x⁻² Multiplying the equation by the I.F. and integrating: y x⁻² = x^2 x x⁻² dx y x⁻² = x dx y x⁻² = x + C y = x^2 x + C x^2 The curve passes through ( , 0) . Substituting x = and y = 0 : 0 = ^2 + C ^2 Since = 0 , we have C = 0 . Thus, the equation of the curve is f(x) = x^2 x . Now, we need to evaluate the definite inte