# 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: $-32 y t^3 = -32 (4 t^4 + 1)^{-1} t^3$.

• After i got this equation: -32yt^3 =-32(4t^4 +1)^-1 t^3 ,what i have to do next – randy Nov 28 '13 at 20:29
• You are asked to estimate this. On what interval? Remember from calculus how to maximize or minimize a function. – Robert Israel Nov 29 '13 at 2:23
• Now i want to plot this partial derivative ∂f/∂y = |-32 (4t^4 +1)^-1 t^3| against t using fortran to generate a file or something for the matlab to plot the graph. Does fortran plot the graph? Help me please – randy Nov 29 '13 at 20:54
• Why in the world would you want to use fortran for that? Matlab can plot it just fine. – Robert Israel Dec 1 '13 at 2:02
• because i have to write program in fortran – randy Dec 1 '13 at 15:13