Can this differential equation be solved analytically or numerically?

[JulieK]

I have the following differential equation

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


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

 

[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.

 

[I like Serena]

I have the following differential equation

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


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

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.

 

[JJacquelin]

The formal solution is in attachment :