Integration of second order partial derivative

[hermano]

Homework Statement

Hi,

I have to solve a boundary condition problem but therefore I have to integrate a second order partial derivative. However, I don't know how to integrate the equation two times. Can someone explain this step by step how I get this solution?

Homework Equations

$\int\frac{\partial^2u}{\partial y^2}= \int\frac{1}{\mu}\frac{\partial p}{\partial x}$

The Attempt at a Solution

$u = \frac{1}{2\mu}( \frac{\partial p}{\partial x} ) y^2 + C1 y + C2$

 

[anathema]

Integrating a second order partial derivative can seem daunting at first, but with some practice and understanding of the process, it can become much easier. Here is a step-by-step guide on how to integrate the equation you have provided:

Step 1: Recognize the pattern
The first step in integrating a second order partial derivative is to recognize the pattern. In this case, we have a second order partial derivative of u with respect to y, which can be written as \frac{\partial^2u}{\partial y^2}. This can also be seen as the second derivative of u with respect to y.

Step 2: Use the chain rule
Next, we can use the chain rule to rewrite the second order partial derivative in terms of x. The chain rule states that the derivative of a function y with respect to x is equal to the derivative of y with respect to u, multiplied by the derivative of u with respect to x.

In this case, we can rewrite the second order partial derivative as \frac{\partial^2u}{\partial y^2} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial y}) = \frac{\partial}{\partial x}(\frac{1}{\mu}\frac{\partial p}{\partial x})

Step 3: Integrate with respect to x
Now that we have rewritten the second order partial derivative in terms of x, we can integrate both sides of the equation with respect to x. This will eliminate the partial derivative with respect to x on the right side of the equation.

\int\frac{\partial^2u}{\partial y^2}dx = \int\frac{1}{\mu}\frac{\partial p}{\partial x}dx

Step 4: Simplify the left side
The left side of the equation can be simplified using the fundamental theorem of calculus, which states that the integral of a derivative is equal to the original function.

\frac{\partial u}{\partial y} = \frac{1}{\mu}\frac{\partial p}{\partial x}x + C1

Step 5: Integrate again
Now that we have eliminated the partial derivative with respect to x, we can integrate both sides of the equation with respect to y.

\int\frac{\partial u}{\partial y}dy = \int\frac{1}{\mu}\frac