# Help with PDE: F(t)g(r)+V/R Derivative

• matteo86bo
In summary, the person is seeking help with a PDE for their thesis that has a physical application. They mention known functions f(t) and g(r) and their interest in how the second term affects the solution when only the first term is used. A suggestion is given to use the Laplace transform method and a general solution is provided.
matteo86bo
I need help with this PDE, it's not an homework, I need to solve it for my thesis and it has physical application...anyway the problem is:
$$\frac{dx}{dt}=f(t)g(r)+\frac{v}{r}\frac{d (Rx)}{dR}$$

$$f(t)$$ and $$g(r)$$ are known.

I can solve the equation with only the first or the second term ...
actually I'm interest in how the second term modify the solution of the equation with the first term only. suggestions?

Your PDE can be solved with help of Laplace transform method. For your purpose it'll be better the following form of general solution ( I assume that in fact R is r)

$$x(t,r) = \frac{1}{r}[\int_c^tf(\xi)g(vt-v\xi+r)(vt-v\xi+r)d\xi+F(vt+r)],$$

where $$F(z)$$ is an arbitrary function, $$c$$ is an arbitrary constant.

Last edited:

## 1. What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple variables and their partial derivatives. It is commonly used in physics and engineering to describe the behavior of systems that change over time.

## 2. What does F(t)g(r) represent in the equation?

F(t)g(r) represents the two functions that are being multiplied together in the equation. F(t) and g(r) can be any mathematical functions, and their product is then used in the equation to describe the relationship between the variables.

## 3. What is the V/R derivative in this equation?

The V/R derivative is a type of derivative that involves taking the derivative with respect to both space (V) and time (R). This type of derivative allows us to analyze how a system changes over both space and time simultaneously.

## 4. How do I solve this PDE?

Solving a PDE involves finding a function that satisfies the equation. This can be done analytically, using mathematical techniques, or numerically, using computer algorithms. The approach used will depend on the specific equation and the desired level of accuracy.

## 5. What are some real-world applications of PDEs?

PDEs have many applications in various fields, such as physics, engineering, economics, and biology. They are commonly used to model physical phenomena, such as heat transfer, fluid flow, and electromagnetic fields. PDEs are also used in finance to describe the behavior of stock prices and in biology to study the growth of populations.

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