1. Given differential equation

$$\sqrt{4-x^2}\,\frac{dy}{dx}=\left(\left(\sin^{-1}\frac{x}{2}\right)^2-y\right)\sin^{-1}\frac{x}{2}$$

with

$$y(2)=\frac{\pi^2-8}{4}.$$

We need to find $y^2(0)$.

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2. Substitute a convenient variable

Let

$$t=\sin^{-1}\left(\frac{x}{2}\right).$$

Then

$$x=2\sin t,$$

so

$$\frac{dx}{dt}=2\cos t.$$

Also,

$$\sqrt{4-x^2}=\sqrt{4-4\sin^2 t}=2\cos t$$

because $t\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, hence $\cos t\ge 0$.

Now,

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{1}{2\cos t}\frac{dy}{dt}.$$

Therefore,

$$\sqrt{4-x^2}\frac{dy}{dx}=(2\cos t)\left(\frac{1}{2\cos t}\frac{dy}{dt}\right)=\frac{dy}{dt}.$$

So the differential equation becomes

$$\frac{dy}{dt}=(t^2-y)t.$$

That is,

$$\frac{dy}{dt}+ty=t^3.$$

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3. Solve the linear differential equation

We have

$$\frac{dy}{dt}+ty=t^3.$$

Its integrating factor is

$$\mathrm{IF}=e^{\int t\,dt}=e^{t^2/2}.$$

Multiplying throughout,

$$e^{t^2/2}\frac{dy}{dt}+te^{t^2/2}y=t^3e^{t^2/2}.$$

Hence,

$$\frac{d}{dt}\left(ye^{t^2/2}\right)=t^3e^{t^2/2}.$$

Integrate both sides:

$$ye^{t^2/2}=\int t^3e^{t^2/2}\,dt+C.$$

Now evaluate the integral. Write

$$t^3=t\cdot t^2.$$

Let

$$u=\frac{t^2}{2}\implies d\nu=t\,dt,$$

and $t^2=2\nu$. Then

$$\int t^3e^{t^2/2}dt=\int t^2e^{t^2/2}(t\,dt)=\int 2\nu e^{\nu}d\nu.$$

Using integration by parts or standard result,

$$\int 2\nu e^{\nu}d\nu=2e^{\nu}(\nu-1)+C.$$

Substituting back $\nu=t^2/2$,

$$\int t^3e^{t^2/2}dt=e^{t^2/2}(t^2-2)+C.$$

Thus,

$$ye^{t^2/2}=e^{t^2/2}(t^2-2)+C,$$

so

$$y=t^2-2+Ce^{-t^2/2}.$$

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4. Use the given condition

When $x=2$,

$$t=\sin^{-1}(1)=\frac{\pi}{2}.$$

Given

$$y\left(2\right)=\frac{\pi^2-8}{4}=\frac{\pi^2}{4}-2.$$

Substitute into the solution:

\frac{\pi^2}{4}-2=\left(\frac{\pi}{2}\right)^2-2+Ce^{-\pi^2/8}=rac{\pi^2}{4}-2+Ce^{-\pi^2/8}.

Hence,

$$Ce^{-\pi^2/8}=0\implies C=0.$$

So,

$$y=t^2-2.$$

Returning to $x$,

$$y(x)=\left(\sin^{-1}\frac{x}{2}\right)^2-2.$$

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5. Find $y(0)$

At $x=0$,

$$\sin^{-1}(0)=0.$$

Therefore,

$$y(0)=0^2-2=-2.$$

So,

$$y^2(0)=(-2)^2=4.$$

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6. Final answer

$$\boxed{4}$$

The derived answer matches the stored correct answer.