I have been given the following:

Consider an anharmonic oscillator which obeys the differential equation

(1)$$x''(t)=3\Omega^2 x^2(t) -4\Omega^2 x^3(t)$$
We are interested in the solution x(t) subject to the initial condition x(0)=1 and x'(0)=0

(i) solve the differential equation within a series expansion

(2)$$x(t)=c_0+c_1t+c_2t^2+c_3t^3+c_4t^4+...$$ and compute the first five coefficients by direct inspection, or otherwise. You may wish to compute first x'''(0) and x''''(0) from (1), and then compare with (2)

(ii) show that: $$x(t)=\frac{1}{1+\frac{1}{2}\Omega^2t^2},$$ is a solution to (1) in closed form.

(iii) Confirm that the Taylor expansion of the exact solution agrees with your result

My Problem

I have done part (i) and (iii), it is (ii) that I am stuck on. I have googled the definition of a closed-form expression, and Wikipedia says that "it is a mathematical expression that can be evaluated in a finite number of operations". This definition has not made it any clearer what the question is asking. Also, I am unsure as to whether the question is asking for me to derive the solution or just to prove that it is a solution, simply by substitution into the series expansion that I have just computed. If anyone has any ideas they could offer, that'd be great.


It just wants you to show that this expression is a solution. You do so by substituting into the differential equation (not the series) and seeing that it works.


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