Partial Differential/Integration Arbitrary Functions

• AntSC
In summary, the solution to the given problem involves using integration to find a function u(x,y,z) involving one or more arbitrary functions. The solution takes the form of u(x,y,z) = x*sinh^-1(y) + f(x,z), which is more general than the form u(x,y,z) = f(x) + g(z).
AntSC
Use integration to find a solution involving one or more arbirary functions
$$\frac{\partial u}{\partial y}=\frac{x}{\sqrt{1+y^2}}$$
for a function $u(x,y,z)$
$$u(x,y,z)=x\int \frac{dy}{\sqrt{1+y^2}}$$
let $y=\sinh v$
$$u(x,y,z)=x\int \frac{\cosh v\: dv}{\sqrt{1+\sinh ^2v}}$$
$$u(x,y,z)=x\sinh ^{-1}y+f(x,z)$$
So here's the question. Why is the solution with an arbirary function $f(x,z)$ and not two arbitrary functions $f(x)+g(z)$? What's the difference?

##f(x,z)## is more general than ##f(x) + g(z)##.

For example, how would you convert a function like ##f(x,z) = x^z## to the form ##f(x) + g(z)##?

AlephZero said:
##f(x,z)## is more general than ##f(x) + g(z)##.

For example, how would you convert a function like ##f(x,z) = x^z## to the form ##f(x) + g(z)##?

Good point. But how can i explain that in a mathematical language?

1. What is a partial differential equation?

A partial differential equation (PDE) is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe physical phenomena that vary in space and time, such as heat transfer, fluid dynamics, and quantum mechanics.

2. What is the difference between a partial derivative and an ordinary derivative?

A partial derivative is a derivative of a function with respect to one of its independent variables, while holding the other variables constant. An ordinary derivative, on the other hand, is a derivative of a function with respect to a single independent variable. Partial derivatives are used in multivariable calculus, while ordinary derivatives are used in single variable calculus.

3. How are partial differential equations solved?

Partial differential equations can be solved analytically or numerically. Analytical solutions involve finding an explicit expression for the solution, while numerical solutions involve approximating the solution using computational methods. The choice of method depends on the complexity of the equation and the desired level of accuracy.

4. What is an integration arbitrary function?

An integration arbitrary function is a function that is added to the general solution of a differential equation to account for any additional boundary conditions. It is called "arbitrary" because it can take any form as long as it satisfies the boundary conditions. This allows for a more specific solution to be obtained for a particular problem.

5. How are partial differential equations used in real-world applications?

Partial differential equations are used in a wide range of real-world applications, including engineering, physics, economics, and biology. They are used to model and analyze complex systems and phenomena, such as the flow of fluids, the spread of diseases, and the behavior of financial markets. Solving PDEs can provide insights and predictions that are crucial for decision-making and problem-solving in various fields.

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