I need to solve an equation of the type, $axe^{qx} + be^{rx} + cx + d = 0$
I tried but couldn't solve it.
Does anyone have an idea how to solve this(for x)?
Thanks
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4The solution is $d=-axe^{qx}-be^{rx}-cx$. Or did you want to solve for $x$? – vadim123 Dec 23 '14 at 05:18
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I want to solve for x. a,b,c,d,q and r are constants. – Manuri Perera Dec 23 '14 at 05:21
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You have to specify the variable to which you want to solve. – David Cardozo Dec 23 '14 at 05:21
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Why do you want to solve this? How would you solve $e^x+x=0$? – David P Dec 23 '14 at 05:21
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2Only numerical methods would do it. If you want me to elaborate, post values of your choice for the different constants. – Claude Leibovici Dec 23 '14 at 05:23
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@ClaudeLeibovici The problem is these constants may take different values. I need a general solution. I also was thinking about a numerical method but then can we get a general solution? – Manuri Perera Dec 23 '14 at 05:32
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Provided that there is one solution, the problem should be very simple with Newton. – Claude Leibovici Dec 23 '14 at 05:34
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Why one solution? Sorry I didn't get it. – Manuri Perera Dec 23 '14 at 05:40
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So an interesting question is then, "Under what conditions does the equation have a unique sokution?" – Robert Lewis Dec 23 '14 at 05:42
1 Answers
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The equation as stated does not admit a symbolic (closed-form) solution. The following variations of it can be solved analytically:
$$ \begin{equation} \begin{split} axe^{qx}+cx & = 0 \quad \text{(Solution: } \frac{1}{q}\log(-\frac{c}{a}) \text{)} \\ axe^{qx}+d & = 0 \quad \text{(Solution: } \frac{1}{q}W(-\frac{dq}{a}) \text{, where W is the Lambert W function)} \\ \end{split} \end{equation} $$
More general forms such as the one you provided are out of reach from an analytical standpoint, but may easily be tackled numerically with any variation of Newton's method, especially since the derivatives can be computed analytically and therefore rapidly evaluated to high precision. Additionally, the optimization function is convex for many parameter choices, so you can expect rapid convergence to a zero without problems.
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So, simply it is not possible to have a general solution for this using any method? – Manuri Perera Dec 23 '14 at 05:57
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@TavionPotter: Not an analytical solution, no. But any sufficiently advanced numerical method will converge to at least one solution (though not necessarily all of them). – Dec 23 '14 at 06:00
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Could you recommend any such method? Sorry I am not very familiar with numerical methods. – Manuri Perera Dec 23 '14 at 06:02
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If you're unfamiliar with numerical equation solving, you'll be best served with the built-in solvers of any computer algebra system (MATLAB, Octave, Mathematica, ...). These systems will apply sophisticated meta-algorithms that are pretty much guaranteed to find not just one but even multiple (all?) solutions. – Dec 23 '14 at 06:07
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Will it be possible to get a solution from them without having specific values for a,b,c.. ? – Manuri Perera Dec 23 '14 at 06:10
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No. Numerical methods require numbers, they cannot work with symbols. You might be able to get an analytical solution when specifying only some of the parameters, though. – Dec 23 '14 at 06:13
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Yeah that is the problem. I'll see what I can do. Thank you for the help. – Manuri Perera Dec 23 '14 at 06:18