# First order non linear Ordinary differential equations

Consider the first order differential equation $\displaystyle\frac{dy}{dt} = f(t,y)= -16t^{3}y^{2}$, with the inital condition $y(0)=1$ Estimate the lipschitz derivative for the differential equation by substituting the exact solution into $\displaystyle\frac{\partial f}{\partial y}$.

I found the exact solution by using the separable of variable and doing integration which is $y(t)= (4t^4 + 1)^{-1}$ And also I found the $\displaystyle\frac{\partial f}{\partial y} = -32yt^3$ The question ask about by substituting the exact solution into $\displaystyle\frac{\partial f}{\partial y}$ to estimate the lipschitz derivative. I don't know how to substitute. Does anyone knows about lipschitz derivative?

I'm not aware of "Lipschitz derivative" as standard terminology in mathematics. However, in the existence and uniqueness theorem for differential equations one of the hypotheses is a Lipschitz condition $|f(t,y) - f(t,z)| \le L |y - z|$, and I believe the question is referring to this $L$. For differentiable functions, it's simply a bound on the partial derivative $\partial f/\partial y$.
"Substitute" means plug in the value of $y(t)$ that you found into the expression for the partial derivative: $-32 y t^3 = -32 (4 t^4 + 1)^{-1} t^3$.