by Darkwizie
Last Updated September 21, 2018 17:20 PM

I have the following set of ordinary differential equations: $$ \begin{cases} x_1'(s) &= x_2(s) - x_1(s) x_3(s)\\ x_2'(s) &= -x_2(s) + x_1(s)^2\\ x_3'(s) &= -x_3(s) + x_1(s), \end{cases} $$ with boundary condition $x_2(0)=x_3(0)=0$ and $x_1(0) = 1/2$.

I am looking for a method to find an exact solution for this type of ODE.

EDIT I have found this link of methods for solving this type of problems, but my problem does not seem to fit in any of the suggested methods.

Following the suggested method we can differentiate the first equation, this way we obtain the equivalent equations: $$ \begin{cases} x_1''(s) &= - x_1'(s) (1+x_3(s))\\ x_3'(s) &= x_1(s) - x_3(s). \end{cases} $$

Just a beginning, you can eliminate $x_3$ $$ \begin{cases} x_1' &= x_2 - x_1 x_3\\ x_2' &= -x_2 + x_1^2\\ x_3' &= -x_3 + x_1, \end{cases} $$

Sum the first and the second: $$x_1'+x_2'=x_1^2-x_1 x_3=x_1(x_1-x_3)$$

So we have

$$ \begin{cases} x_3' &= {x_1' +x_2' \over x_1} \\ x_3 &= x_1 - {x_1' +x_2' \over x_1} , \end{cases} $$

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