The solution to the differential equation d y d x = y f^ (x)-y² f(x) where f(x) is a given function is

Step-by-step solution

We have d y d x = f^ (x) f(x) y- y² f(x) d y d x - f^ (x) f(x) y=- y² f(x) Divide by y²: y⁻² d y d x -y⁻¹ f^ (x) f(x) =- 1 f(x) Put y⁻¹=z -y⁻² d y d x = d z d x - d z d x - f^ (x) f(x) (z)=- 1 f(x) d z d x + f^ (x) f(x) z= 1 f(x) I F=e^ f^ (x) f(x) d x =e^ f(x) =f(x) The solution is z(f(x))= 1 f(x) (f(x)) d x+c y⁻¹(f(x))=x+c f(x)=y(x+c)