Integral Equation to Differential

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Problem

Say I have the following equation.

$y=f(\theta)$

Where

$\theta = \int\int\alpha$

Is it possible to express the equation in terms of $\alpha$ and not the double integral of it?

Origin

This problem originates from the problem shown in the image below.

Problem.jpg

Where the equations $F_y=M_ysin(\frac{\pi}{2}-\theta)$ and $F_x=M_ycos(\frac{\pi}{2}-\theta)$ both depend on the angle $\theta$ but where only the angular acceleration $\alpha$ is known.

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With the additional comments, the angular acceleration $\alpha$ is a constant (not depending of $\theta$). You can then write $$\theta=\theta_0+\omega_0t+\frac 12\alpha t^2$$ Here $\theta_0$ is the original angle of the rocket, and $\omega_0$ is the original angular velocity. $t$ is time.