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I came across a system of differential equations in the form: $\newcommand{\D}[1]{\frac{\mathrm{d}#1}{\mathrm{d}x}}$ \begin{align} f_1(x,y,z)\D{y}+f_2(x,y,z)\D{z}&=f_3(x,y,z),\\ f_4(x,y,z)\D{y}+f_5(x,y,z)\D{z}&=f_6(x,y,z). \end{align}

I numerically solved the system as follows. Let $\Phi$ be given by $$ \Phi(x,y,z)=\begin{bmatrix}y'\\z'\end{bmatrix}= \begin{bmatrix} f_1(x,y,z)&f_2(x,y,z)\\ f_4(x,y,z)&f_5(x,y,z) \end{bmatrix}^{-1}\begin{bmatrix} f_3(x,y,z)\\ f_6(x,y,z) \end{bmatrix}, $$

then Euler's method takes the form $$ \begin{bmatrix} y\\ z \end{bmatrix}_{(i+1)}= \begin{bmatrix} y\\ z \end{bmatrix}_{(i)} +h~\Phi(x,y,z). $$

I tested this approach (using RK4 instead of Euler) for simple systems (for which I can obtain analytical solutions) and it seems to work.

Other than the matrix of $\{f_1,f_2,f_4,f_5\}$ being singular, are there any other conditions for which this method fails?

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  • $\begingroup$ Generally speaking, any numerical integration method will fail if either your problem is ill-posed (you supply the wrong number/type of boundary condition or impose it/them at the wrong place), or if it is not stable (your Euler or RK step is too large). $\endgroup$ – ekkilop May 30 '16 at 11:18

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