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Re: Integration by substitution.
Posted:
Dec 17, 2012 7:07 AM


> I'm sorry, I try to better explain my point. > Let's suppose this is a ODEs I want to solve. > http://www.mathworks.it/help/matlab/ref/eqn1311792846. > png > > y' are derivatives w.r.t the time. > I could use a RungeKutta method to solve them from > t0 a the final time (please, see > http://www.mathworks.it/it/help/matlab/ref/ode45.html) > . Well done. > > Now let's suppose I would integrate my problem in > auxiliary variable, let' say E. It is only > theoretically. > (e.g. E = t^2 or E = sin(t),...). I am able to expres > t0 e in tf in terms of E. > How can I solve my ODE now? It is sufficient to > multiply y' with dt/dE and use my Runge Kutta? > > Thanks for your support
Let's say t~ = g(t). Then dt~/dt = d/dt (g(t)) and thus  if y~(t~) = y(t)  : dy/dt = dy~/dt~ * dt~/dt = dy~/dt~ * d/dt(g(t)) = dy~/dt~ * (d/dt(g(t))_evaluated at t=g^(1)(t~)). Thus you will have to solve the following ODE in t~: dy~/dt~ = f(y~(t~))/((d/dt(g(t))_evaluated at g^(1)(t~)).
E.g. if t~ = t^2, dy/dt = f(y) transfroms to dy~/dt~ = f(y~)/(2*sqrt(t~))
Best wishes Torsten.



