How does one find eigenvalues $\lambda$ of the following problem? $$ \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} = \lambda \left( -u + u^2 \right), $$ $$ u(0) = u(1) = 0. $$ Can this problem be tackled easily by some software? I have looked into Mathematica but its DEigensystem method is unfortunately taylored for the right hand side of the form $\lambda u$ only.
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$\begingroup$ Maybe substitute to complete a square $\endgroup$– mathreadlerMay 30, 2018 at 14:50
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$\begingroup$ @mathreadler Will that help? I will still have to deal with the absolute term $\lambda/4$ which pops up this way. $\endgroup$– sleepingrabbitMay 31, 2018 at 6:16
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$\begingroup$ If you complete a square maybe you can do a subsequent substitution. $\endgroup$– mathreadlerMay 31, 2018 at 6:18
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$\begingroup$ @mathreadler If you show me how it can be done to obtain $\mu v$ on the right hand side for some new variables $\mu$, $v$, I will accept that as an answer. $\endgroup$– sleepingrabbitMay 31, 2018 at 6:28
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1 Answer
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A satisfactory answer (to me) appeared on Mathematica StackExchange:
https://mathematica.stackexchange.com/questions/174346/nonlinear-ode-eigenvalue-problem