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I have been reading the Strogatz book on Nonlinear Ordinary Differential equations and I understand the graphical/qualitative method to solving these types of equations. However, Strogatz did not seem to address the role of numerical methods in solving nonlinear ODEs or systems of ODEs. I also looked at the book by Jordan and Smith on nonlinear ODEs, but they don't have a chapter on numerical methods for ODEs either.

So I was just trying to understand what is the role of numerical methods in looking at nonlinear odes? The basic approach of the qualitative methods seems to be curve sketching, so in a sense we are already using a numerical method--but we are drawing the nonlinear ODE itself instead of its integral curves right. But how does this approach relate to using numerical methods for linear ODEs like Euler, or Runge-Kutta, or Tsit?

UPDATE:

Tsit stands for Tsitouras-5 ODE solver.

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Basically you use numerical methods when you want (a good approximation of) an actual value $y(x)$. The qualitative methods might give you a very rough ballpark estimate, but numerical methods will give you the actual numbers.

I'm a bit confused by your writing "numerical methods for linear ODE's". Numerical methods such as Euler and Rung-Kutta work just as well for nonlinear ODE's as for linear ones. And what is "Tsit"?

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  • $\begingroup$ thanks for the input. Sorry, by Tsit I meant the Tsitouras-5 ode solver, which I think is an adaptation of RK-4. Okay, so Euler and RK work fine for both linear and nonlinear ODEs then, so that clarifies that point of confusion. So if I need an exact solution I can use one of those methods. By "ballpark" do you mean that the qualitative methods let me sketch the direction field but don't give me an exact solution. $\endgroup$ – krishnab Jul 24 at 23:34

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