Solving DE- which approach and more importantly - why

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In summary, the conversation discusses different methods for solving the differential equation \frac{d}{dm}[F(m) G(m)] = (\frac{d}{dm}F(m)) D(m). The first method involves using integration by parts, while the second method involves a separation of variables. The conversation also mentions the need for specifying functions D and G in order to evaluate the right-hand side of the equation. Ultimately, the conversation concludes with a suggestion to integrate both sides with respect to m.
  • #1
CJDW
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My end goal is to solve for [itex]G(m)[/itex] in terms of the other functions, but first I have to solve the DE :

[itex]\frac{d}{dm}[F(m) G(m)] = (\frac{d}{dm}F(m)) D(m)[/itex].

What I've done is to say (using integration by parts)

[itex]F(m) G(m) = F(m)D(m) - \int F(m) (\frac{d}{dm}D(m)) dm[/itex].

This is one method I tried. Another approach would be a separation of variables, which I tried as...

[itex]\frac{d}{dm}[F(m) G(m)] = \frac{dF}{dm} G + F \frac{dG}{dm} = \frac{dF}{dm} D [/itex]

which leads to...

[itex] \ln |F| = \int \frac{dG}{D - G} dm [/itex] and I believe that the RHS cannot be evaluated without specifying functions [itex]D[/itex] and [itex]G[/itex] (if the RHS can be evaluated without specification - please let me know...and please show me the way).

Help on understanding the correct method would be extremely appreciated.

Thanks in advance.
 
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  • #2
If you just need an expression for G in terms of F and D, then the first one looks more like it gets you there.

I noticed that $$\renewcommand{\dd}[2]{\frac{d #1}{d #2}}
\renewcommand{\dm}[1]{\frac{d #1}{dm}}

G\dm{F}+ F\dm{G} = D\dm{F}$$ rearranges to $$\dm{G}= \frac{D-G}{F}\dm{F}$$ ... integrate both sides wrt m ... but it's early and I havn't had my coffee yet...
 
  • #3
Hi Simon, thanks for your reply.

Thanks for your suggestion, but I think I'll stick with the integration by parts method, since I have convenient expressions for [itex]F(m) (\frac{d}{dm}D(m))[/itex] that I will need to integrate.

Once again, thanks for the help.
 
  • #4
Hi !
it's a separable EDO :
 

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  • #5
Yeah - that's where I was going ;)
(Once I'd had my coffee...)
 

1. What is the best approach for solving differential equations?

The best approach for solving differential equations depends on the type of differential equation and the available tools. Some common approaches include separation of variables, using a substitution method, and using an integrating factor. Each method has its own advantages and disadvantages, so it is important to consider the specific equation and choose the most suitable approach.

2. Why is it important to choose the right approach for solving differential equations?

Choosing the right approach for solving differential equations ensures that the solution is accurate and efficient. A wrong approach may lead to incorrect solutions or may require more time and effort to solve the equation. Additionally, understanding the different approaches helps in developing a deeper understanding of the concepts and principles behind solving differential equations.

3. Can differential equations be solved analytically or numerically?

Differential equations can be solved using both analytical and numerical methods. Analytical methods involve finding an exact solution by manipulating the equation algebraically, while numerical methods involve approximating the solution using numerical techniques. Both methods have their own advantages and are used depending on the complexity of the equation and the desired level of accuracy.

4. What are some common tools used for solving differential equations?

Some common tools used for solving differential equations include calculus techniques, such as integration and differentiation, linear algebra, and numerical methods like Euler's method and Runge-Kutta methods. Software programs like MATLAB and Mathematica are also commonly used for solving differential equations.

5. How can solving differential equations be applied in real-world problems?

Differential equations are used to model and understand various real-world phenomena, such as population growth, chemical reactions, and electrical circuits. By solving differential equations, we can predict and analyze the behavior of these systems and make informed decisions. This makes differential equations an essential tool in various fields, including physics, engineering, economics, and biology.

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