# Solving differential equation

1. Mar 4, 2013

### JulieK

I have the following differential equation

\frac{\partial b}{\partial x}=\frac{b-c}{c^{2}}

where $b$ and $c$ are both functions of $x$. However, although
I have a closed form relation between $c$ and $x$, I do not have
such a closed form relation between $b$ and $x$. Is there any analytic
or numeric way to solve this problem. I want a solution linking $b$,
$c$ and $x$

2. Mar 4, 2013

### bigfooted

Well, when c(x) is a known (and integrable) function, then the ode for b(x) is a linear first order ODE and you can solve it by finding the integrating factor.

3. Mar 4, 2013

### I like Serena

Hi JulieK!

Wolfram|Alpha gives this analytic solution.
To get a nicer formula, you need a specific c(x).

To solve numerically, the simplest method you can use is Euler's method.
Euler's method uses that:
$$db=\frac{b-c}{c^{2}}dx$$
From a given $x_0$ and $b_0$, and with a stepsize $h$, the algorithm is:
\left[ \begin{align}x_{n+1} &= x_n + h \\ b_{n+1} &= b_n + h \frac{b_n-c(x_n)}{c(x_n)^{2}} \end{align} \right.

A more advanced and accurate method is Runge-Kutta, which is described here.

4. Mar 5, 2013

### JJacquelin

The formal solution is in attachment :

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