Solving differential equation

  • Thread starter JulieK
  • Start date
  • #1
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I have the following differential equation

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


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

Answers and Replies

  • #2
bigfooted
Gold Member
617
134
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
I like Serena
Homework Helper
6,579
176
I have the following differential equation

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


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

Hi JulieK! :smile:

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
798
34
The formal solution is in attachment :
 

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