Thread: How to integrate this partial differential equation

1. How to integrate this partial differential equation

I have the following equation

$\displaystyle \frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0$

where $\displaystyle y$ is a function of $\displaystyle x$ and $\displaystyle m$ is a function of $\displaystyle y$. If I integrate this equation first with respect to $\displaystyle y$ should I get a function of $\displaystyle x$ as the constant of integration (say $\displaystyle C\left(x\right)$) or it is just a constant? If it is a function, how can I then find its form (e.g. polynomial, etc.)? Should I use boundary conditions or I can decide about the form from inspecting the type of the equation.

2. Re: How to integrate this partial differential equation

Originally Posted by JulieK
I have the following equation

$\displaystyle \frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0$

where $\displaystyle y$ is a function of $\displaystyle x$ and $\displaystyle m$ is a function of $\displaystyle y$. If I integrate this equation first with respect to $\displaystyle y$ should I get a function of $\displaystyle x$ as the constant of integration (say $\displaystyle C\left(x\right)$) or it is just a constant? If it is a function, how can I then find its form (e.g. polynomial, etc.)? Should I use boundary conditions or I can decide about the form from inspecting the type of the equation.
\displaystyle \displaystyle \begin{align*} \frac{\partial }{\partial y} \left( m\, \frac{dy}{dx} \right) &= 0 \\ m\,\frac{dy}{dx} &= \int{0\, dy} \\ m\,\frac{dy}{dx} &= f(x) \\ \frac{dy}{dx} &= \frac{f(x)}{m} \\ y &= \int{\frac{f(x)}{m}\,dx} \\ y &= \frac{1}{m} \int{f(x)\,dx} + C \end{align*}

Unless you have some boundary conditions now, this is the simplest it's going to get.