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I am working on a physics problem that is probably a bit too difficult for me since I am not adept at calculus. I am solving a differential equation and I'm stuck at this step since I only know how to take simple derivatives and wolfram alpha cannot process this specific equation:

$$\int_0^t \ dt' = \int^v_0\frac{dV'}{A_N\sqrt{\frac{2\left(\left(\frac{V_0}{V'}\right)^k-P_a\right)}{p_w}}}$$

All of the variables are constants except for V and t. Can this be evaluated without too much of a hassle? If so, what is the solution and if not, where can I look to find out how to solve this type of equation?

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  • $\begingroup$ This equation doesn't seem to make sense. On the right side you are integrating with respect to $t$ but $t$ also appears in the limits of integration. $\endgroup$ – Biggs Oct 6 '16 at 4:35
  • $\begingroup$ Sorry, I edited the question, does this help at all? @Biggs $\endgroup$ – Ryan Oct 6 '16 at 4:38
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This is not an answer since the result was obtained just using a CAS.

This is one of the typical cases where appears the gaussian hypergeometric function.

$$ \int\frac{dV}{A\sqrt{\frac{2\left(\left(\frac{v}{V}\right)^k-P\right)}{p}}}=-\frac{p V \sqrt{\frac{\left(\frac{v}{V}\right)^k-P}{p}}}{\sqrt{2} A P}\,\,\, _2F_1\left(1,\frac{1}{2}-\frac{1}{k};\frac{k-1}{k};\frac{\left(\frac{v}{V}\right )^k}{P}\right) $$ making $$ \int^v_0\frac{dV}{A\sqrt{\frac{2\left(\left(\frac{v}{V}\right)^k-P\right)}{p}}}=\frac{v }{\sqrt{2} A \sqrt{-\frac{P}{p}}}\,\,\, _2F_1\left(\frac{1}{2},-\frac{1}{k};\frac{k-1}{k};\frac{1}{P}\right)$$ provided $\Re(k)<0\land \Re\left(\frac{P}{p}\right)<0\land v>0$.

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