Editing Appendix/Ramblings/T3Integrals/QuadraticCase (section)

===Solution Strategy===
I'm not sure whether the following strategy is fully legitimate, but let's explore it anyway.  Because the LHS of [[Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01|'''Equation T3Q.01''']] displays an explicit dependence only on the coordinate <math>\lambda_2</math> while the RHS displays an explicit dependence only on <math>\Lambda</math> &#8212; that is, only on the ''ratio'' of the two coordinates <math>\lambda_1/\lambda_2</math> &#8212; perhaps we can use a ''separation of variables'' technique to derive a solution.  Specifically, suppose the LHS and the RHS separately are set equal to the same value, call it <math>n</math>. 

Then, for the LHS:
<div align="center">
<math>
\ddot{\lambda}_2 = \frac{n}{2}\lambda_2 ;
</math>
</div>

And, for the RHS:
<div align="center">
<math>
\frac{d\ln(\Lambda-1)}{dt} = \sqrt{n} .
</math>
</div>
Now I suppose that, in general, <math>n</math> should be allowed to vary with time, but for exploratory purposes, let's assume that <math>n</math> is a constant.  The solution to the LHS's <math>2^\mathrm{nd}</math>-order ODE is,
<div align="center">
<math>
\lambda_2 = \lambda^0_2 \exp{[-\sqrt{n/2}~t]} ,
</math>
</div>
where, <math>\lambda^0_2</math> is the coordinate position <math>\lambda_2</math> at time <math>t=0</math>.  The solution to the RHS's <math>1^\mathrm{st}</math>-order ODE is,
<div align="center">
<math>
\sqrt{n} t = \ln\biggl( \frac{\Lambda-1}{\Lambda_0 -1} \biggr) ,
</math>
</div>
where, <math>\Lambda_0</math> is given by the coordinate ratio at time <math>t=0</math>, specifically, <math>\Lambda_0 \equiv \sqrt{1 + (2\lambda_1^0/\lambda_2^0)^2}</math>.

Now, if we replace "<math>\sqrt{n}~t</math>" in the first of these expressions by the second expression, we find,
<div align="center">
<math>
\frac{\lambda_2}{\lambda^0_2} = \exp{\biggl[ -\frac{1}{\sqrt{2}} \ln\biggl( \frac{\Lambda-1}{\Lambda_0 -1} \biggr) \biggr]} = \biggl( \frac{\Lambda_0 -1}{\Lambda-1} \biggr)^{1/\sqrt{2}} .
</math>
</div>
This would be a fantastically simple result, if it proved to be a proper solution to the governing equation of motion.  Unfortunately, if this relatively elementary equation is differentiated twice in an effort to reproduce [[Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01|'''Equation T3Q.01''']], we find that an additional undesirable term appears that involves the second derivative of a function containing the variable <math>\Lambda</math>.  It can be shown that this undesirable term goes to zero if <math>\sqrt{n}</math> is assumed to be independent of time (as we did indeed assume, above).  Unfortunately, in reality, this does not seem to be a desirable assumption for the physical problem in which we have interest, so we must conclude that the derived elementary equation is not the desired solution of the equation of motion.

But can we learn something valuable from this failed separation of variables approach???