Finding Solutions to $\frac{\partial^2 V}{\partial u \partial v} = 0$

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In summary, the equation $\frac{\partial^2 V}{\partial u \partial v} = 0$ is a partial differential equation used in physics and engineering to find solutions for physical systems. It can be solved using various mathematical methods, and has applications in fields such as heat transfer, fluid flow, electromagnetism, and economics. This equation is also closely related to the concept of equilibrium, as it is often used to analyze and predict the behavior of systems in balance or stability.
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Given [tex]\frac{\partial^2 V}{\partial u \partial v} = 0[/tex], the solution is [tex]V_1(u) + V_2(v)[/tex] for arbitrary [tex]V_1 , V_2[/tex].

I solve to get [tex]\frac{\partial V}{\partial u} = V_3(u)[/tex] and then [tex]V = \int V_3(u) du + C[/tex] Where V_3, C are arbitrary.

How could I transform my latter solution into the first solution? Don't V_1, V_2 have to have some properties such as differentiability? (I found this in a physics textbook)
 
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  • #2
try the same method but integrate over the other variable first then equate the two results
 

FAQ: Finding Solutions to $\frac{\partial^2 V}{\partial u \partial v} = 0$

1. What is the meaning of the equation $\frac{\partial^2 V}{\partial u \partial v} = 0$?

The equation $\frac{\partial^2 V}{\partial u \partial v} = 0$ is a partial differential equation that represents a surface where the rate of change of a function V with respect to two different variables, u and v, is equal to zero. This equation is often used in physics and engineering to find solutions for different physical systems.

2. What type of problems can be solved using this equation?

This equation can be used to find solutions for a wide range of physical systems, including heat transfer, fluid flow, electromagnetism, and elasticity. It is especially useful in problems involving two or more independent variables.

3. How is this equation solved?

The solution to this equation involves finding a function V(u,v) that satisfies the equation $\frac{\partial^2 V}{\partial u \partial v} = 0$. This can be done using various mathematical methods, such as separation of variables, integration, or series expansions. The specific method used will depend on the problem at hand.

4. What are some real-world applications of this equation?

This equation has many practical applications in fields such as physics, engineering, and economics. It can be used to model and solve problems related to heat transfer in buildings, fluid flow in pipes and channels, electrical circuits, and financial markets.

5. How does this equation relate to the concept of equilibrium?

The equation $\frac{\partial^2 V}{\partial u \partial v} = 0$ is often used to find solutions for physical systems in equilibrium. This is because when the rate of change of a function V with respect to two variables is equal to zero, it means that the system is in a state of balance or stability. This makes the equation useful in analyzing and predicting the behavior of systems in equilibrium.

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