# Nonlinear ODE eigenvalue problem

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.

• Maybe substitute to complete a square – mathreadler May 30 '18 at 14:50
• @mathreadler Will that help? I will still have to deal with the absolute term $\lambda/4$ which pops up this way. – sleepingrabbit May 31 '18 at 6:16
• If you complete a square maybe you can do a subsequent substitution. – mathreadler May 31 '18 at 6:18
• @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. – sleepingrabbit May 31 '18 at 6:28