Calculus · Medium problem

Consider the differential equation $\dfrac{dy}{dx}=2xy^2$.

• Write an equation for the line tangent to the solution curve through $(0,1)$, and use it to approximate $f(0.1)$, where $y=f(x)$ is that solution.
• Find the particular solution $y=f(x)$ to the differential equation with initial condition $f(0)=1$.
• State the largest open interval containing $x=0$ on which the solution in part (b) is defined.