ODE Methods for Physicists (related question)

In summary, the conversation is about a differential equation that involves the derivative of a function and its power. The solution to the equation involves using an integrating factor and can be found on a tutorial website. Additionally, it is clarified that aerodynamic force varies with the square of velocity, not the cube.
  • #1
profgabs05
4
2
Homework Statement
A mass 𝑚 is accelerated by a time-varying force 𝛼 𝑒𝑥𝑝(−𝛽𝑡)𝑣3, where v is its velocity. It also experiences a resistive force 𝜂𝑣, where 𝜂 is a constant, owing to its motion through the air. The equation of motion of the mass is therefore
𝑚𝑑𝑣/𝑑𝑡= 𝛼 𝑒𝑥𝑝(−𝛽𝑡)𝑣^3 − 𝜂𝑣.
Find an expression for the velocity v of the mass as a function of time, given that it has an initial velocity 𝑣0
Relevant Equations
𝑚𝑑𝑣/𝑑𝑡= 𝛼 𝑒𝑥𝑝(−𝛽𝑡)𝑣^3 − 𝜂𝑣.
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  • #2
$$\frac{1}{v^3}\frac{dv}{dt}=-\frac{1}{2}\frac{dv^{-2}}{dt}$$
 
  • #3
Please can i get a working guide to this answer
Chestermiller said:
$$\frac{1}{v^3}\frac{dv}{dt}=-\frac{1}{2}\frac{dv^{-2}}{dt}$$
Please can i get a working guide to this answer?
 
  • #4
profgabs05 said:
Please can i get a working guide to this answer
Please can i get a working guide to this answer?
$$\frac{dx^n}{dx}=nx^{n-1}$$
 
  • #5
Last edited:

Related to ODE Methods for Physicists (related question)

1. What are ODE methods and how are they used in physics?

ODE methods, or ordinary differential equation methods, are mathematical techniques used to solve equations that describe the relationship between a function and its derivatives. In physics, ODE methods are commonly used to model and analyze the behavior of physical systems, such as the motion of objects under the influence of forces or the change in temperature over time.

2. What are some common ODE methods used by physicists?

Some common ODE methods used by physicists include Euler's method, Runge-Kutta methods, and the shooting method. These methods vary in complexity and accuracy, and are chosen based on the specific problem being solved.

3. How do ODE methods differ from PDE methods?

ODE methods are used to solve ordinary differential equations, which involve only one independent variable. PDE methods, on the other hand, are used to solve partial differential equations, which involve multiple independent variables. PDE methods are typically more complex and require advanced mathematical techniques.

4. Can ODE methods be used to solve real-world problems?

Yes, ODE methods are commonly used in physics to solve real-world problems. They have been successfully applied to a wide range of fields, including mechanics, thermodynamics, electromagnetism, and quantum mechanics.

5. What are some challenges associated with using ODE methods?

One challenge with using ODE methods is that they may not always provide an exact solution to a problem, but rather an approximation. The accuracy of the approximation depends on the complexity of the problem and the chosen method. Another challenge is that some problems may require a combination of different methods to accurately model the behavior of a system.

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