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I have an autonomous system $y''= \exp(y)$ with initial conditions $y(0) = 1, y'(0) = \sqrt{2 e}$, which I have to solve numerically by secod-order RG method. (Actuall I must solve BVP, but now i'm interested in another question).

I have recurrent formulas for y (as for second first-order equation): $y(t+h) = y(t) + \frac{1}{2} \left(F_1+F_2\right)$ where $F_1 = h \exp(x,y)$ and $F_2 = h \exp(x+h, y+F_1)$. I have the initial value for $F_1$ to start a procedure, but I don't understand how to calculate $F_2$ in my case. I mean what should I do to use or not to use $x+h$ in calculating $F_2$?

Please help me. Thank you very much.

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  • $\begingroup$ Yes, I did so and got an analytic implicit solution which seems to be right (my teacher said and Kamke's book if you know). And it looks horrible as for me, you're right. $\endgroup$ – Seva Dec 26 '14 at 3:30
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Not sure where the formulas in the question come from but the obvious reference suggests to rewrite the second-order differential equation as the first-order differential system $$z'=f(z),\qquad z=(y',y),\qquad f:(u,v)\mapsto(\mathrm e^v,u),$$ hence RK2 applied to $z$ reads $$z \to z+hf(z+\tfrac12hf(z)),$$ or, equivalently, $$y'\to y'+h\mathrm e^{y'+hy/2},\qquad y\to y+h\left(y'+\tfrac12h\mathrm e^y\right).$$ Finally, RK2 amounts to iterating the pair of relations $$y'(t+h)=y'(t)+h\mathrm e^{y'(t)+hy(t)/2},\quad y(t+h)=y(t)+h\left(y'(t)+\tfrac12h\mathrm e^{y(t)}\right).$$

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