Study guide: Solving nonlinear ODE and PDE problems

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Exact solution of quadratic nonlinear equations

Solution of $ F(u)=0 $: $$ u = \frac{1}{2\Delta t} \left(-1+\Delta t \pm \sqrt{(1-\Delta t)^2 - 4\Delta t u^{(1)}}\right) $$

Observation:

Nonlinear algebraic equations may have multiple solutions!