Ordinary differential equations

In summary, the conversation is about estimating the Lipschitz derivative for a first order differential equation with the given initial condition. The exact solution is found by using the separable of variable and doing integration, and the Lipschitz derivative is calculated by substituting the exact solution into ∂f/∂y. The conversation also includes a request for help in finding information about Lipschitz derivative.
  • #1
ra_forever8
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Consider the first order differential equation
dy/dt = f(t,y)= -16t^3y^2, with the inital condition y(0)=1.
Estimate the lipschitz derivative for the differential equation by substituting the exact solution into ∂f/∂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 ∂f/∂y = -32yt^3
The question ask about by substituting the exact solution into ∂f/∂y to estimate the lipschitz derivative. I don't know how to substitute.
Does anyone knows about lipschitz derivative?

Help me Please.
 
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  • #2
ra_forever8 said:
Consider the first order differential equation
dy/dt = f(t,y)= -16t^3y^2, with the inital condition y(0)=1.
Estimate the lipschitz derivative for the differential equation by substituting the exact solution into ∂f/∂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 ∂f/∂y = -32yt^3
The question ask about by substituting the exact solution into ∂f/∂y to estimate the lipschitz derivative. I don't know how to substitute.
Does anyone knows about lipschitz derivative?

Help me Please.

1) Read your textbook; or
2) Google 'lipschitz derivative'.
 
  • #3
I can not find it. So, i was asking for help.
 

FAQ: Ordinary differential equations

What are ordinary differential equations?

Ordinary differential equations (ODEs) are mathematical equations that describe the relationship between a dependent variable and one or more independent variables, where the dependent variable is a function of the independent variables and their derivatives.

What are some real-world applications of ordinary differential equations?

ODEs are used to model many natural phenomena, such as population growth, chemical reactions, and electric circuits. They are also commonly used in engineering fields, such as in the analysis of mechanical systems and control systems.

How do you solve ordinary differential equations?

There are various techniques for solving ODEs, including separation of variables, substitution, and using power series. The method used depends on the type of ODE and its complexity.

What is the difference between ordinary differential equations and partial differential equations?

The main difference between ordinary differential equations and partial differential equations is that ODEs involve only one independent variable, while PDEs involve multiple independent variables. This makes PDEs much more challenging to solve, as they require more advanced mathematical techniques.

Why are ordinary differential equations important?

ODEs are essential in many scientific and engineering fields for modeling and predicting the behavior of systems. They also have numerous practical applications in fields such as medicine, economics, and biology. Additionally, many mathematical concepts and techniques are based on ODEs, making them fundamental in the study of mathematics.

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