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Cannot solve this algebraic equation in Maxima?

Hello i am trying to solve a algebric equation in maxima, the equation has alpha, delta and psi as variable. I want the alpha in equation to be solved in terms of psi and delta. I tried using the solve command but i am getting alpha in terms of alpha.

Here is the equation to solve Equation to solve

And this is the output from maxima

.

This is the code i am trying -->

solve([(sqrt(-4*alpha*delta*psi-4*delta*psi+alpha^2*delta^2)/(delta^2+delta)-(alpha*delta/(delta^2+delta)))/2-sqrt(4*alpha^2*delta^2+6*alpha*delta^2+3*delta^2+2*alpha^2*delta+4*alpha*delta+2*delta)/(3*delta^2+2*delta)+alpha*delta/(3*delta^2+2*delta)=0],alpha);

Thank you

Solution

  • The problem with that algebraic equation is that involves square roots,or radicals and normal polynomial, and that that type of equation is not easy to solve, take a look at this equation:

    (%i30) solve(x=sqrt(x+6),x);

(%o30) x = sqrt{x+6}

    So Maxima doesn't return any value, but for example other software Mathematica does. Let's square both sides of equation and try to solve it

    (%i31) solve(x^2=x+6,x);

(%o31) x=3 , x=-2

    we get two solutions, let's try with first equation:

    3 = sqrt(3+6) => 3 = sqrt(9) => 3 = 3

-2 = sqrt(-2+6) => -2 = sqrt(4) => -2 = 2 ??????

    So the second solution is not valid,

    maxima solve program in Macsyma/Maxima generally avoids methods that produce false solutions, like "square both sides". It may still make errors based on dividing by expressions that appear to be non-zero, but actually ARE zero, and maybe some other similar situations.

    from this mailing list

    In your case I will factor the equation to get a simplified version but with those free variable this will be difficult so, try to assume some values for psi and delta:

    (%i26) solve(factor((sqrt(-4*alpha*delta*psi-4*delta*psi+alpha^2*delta^2)/(delta^2+delta)-(alpha*delta/(delta^2+delta)))/2-sqrt(4*alpha^2*delta^2+6*alpha*delta^2+3*delta^2+2*alpha^2*delta+4*alpha*delta+2*delta)/(3*delta^2+2*delta)+alpha*delta/(3*delta^2+2*delta))=0,alpha);

(\%o26) \left[ \alpha=\ifrac{\left(3\,\delta+2\right)\,\isqrt{\left(-4\,\alpha-4\right)\,\delta\,\psi+\alpha^2\,\delta^2}+\left(-2\,\delta-2\right)\,\isqrt{\left(4\,\alpha^2+6\,\alpha+3\right)\,\delta^2+\left(2\,\alpha^2+4\,\alpha+2\right)\,\delta}}{\delta^2} \right]

    Expand your Equation and try to remove square roots or some assumptions:

    Equation : (sqrt(-4*alpha*delta*psi-4*delta*psi+alpha^2*delta^2)/(delta^2+delta)-(alpha*delta/(delta^2+delta)))/2-sqrt(4*alpha^2*delta^2+6*alpha*delta^2+3*delta^2+2*alpha^2*delta+4*alpha*delta+2*delta)/(3*delta^2+2*delta)+alpha*delta/(3*delta^2+2*delta);