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 :

 

[blue_raver22]

.I can suggest several approaches to solving this differential equation. One option is to use numerical methods such as Euler's method or Runge-Kutta methods to approximate a solution. These methods involve breaking down the equation into small steps and using iterative calculations to approximate the values of b and c at each step. While this may not provide an exact solution, it can give a good approximation and can be useful in situations where a closed form solution is not possible.

Another option is to use analytical methods, such as separation of variables or the method of integrating factors, to try and find an exact solution. These methods can be more complex and may require a deeper understanding of differential equations, but they can provide a precise solution that links b, c, and x.

In some cases, it may also be helpful to use computer software or programming languages to solve the differential equation. This can be particularly useful if the equation is complex or if there are multiple variables involved. With the help of technology, it is possible to find solutions that would be difficult or impossible to obtain by hand.

Overall, the best approach to solving this differential equation will depend on the specific context and the desired level of accuracy. As a scientist, it is important to carefully consider the available methods and choose the most appropriate one for the situation at hand.