Differential Equation - Finding the Particluar Integral

TPMWITAM
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I'm working on this problem and have found the complimentary function, but I'm not quite sure I understand how to get the particular integral from the complimentary function. The textbook I have makes it seem as though it is pretty much guess work, but I was wondering if there is any sort of method that is better than just guessing. My work is scanned and attached as an image ;)

Thanks!
 

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I think you need to consider the t>0 and t<0 cases separately because of the absolute value in the forcing function.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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