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- #1

Given the following ODE

\[

\left(\frac{du}{dx}\right)^2 = \mu u^2 - \frac{2\alpha}{\sigma + 2}u^{\sigma + 2} - \frac{\gamma}{\sigma + 1}u^{2(\sigma + 1)}

\]

How do I obtain

\[

u(x) = \left(\frac{A}{B + \cosh(Dx)}\right)^{1/\sigma}

\]

where

\(A = \frac{(2 + \sigma)B\mu}{\alpha}\), \(B = \text{sgn}(\alpha)\left[1 + \frac{(2 + \sigma)^2\gamma}{(1 + \sigma)\alpha^2}\mu\right]^{-1/2}\), and \(D = \sigma\sqrt{\mu}\) with the variable transformation \(y = u^{-\sigma}\)?

\[

\left(\frac{du}{dx}\right)^2 = \mu u^2 - \frac{2\alpha}{\sigma + 2}u^{\sigma + 2} - \frac{\gamma}{\sigma + 1}u^{2(\sigma + 1)}

\]

How do I obtain

\[

u(x) = \left(\frac{A}{B + \cosh(Dx)}\right)^{1/\sigma}

\]

where

\(A = \frac{(2 + \sigma)B\mu}{\alpha}\), \(B = \text{sgn}(\alpha)\left[1 + \frac{(2 + \sigma)^2\gamma}{(1 + \sigma)\alpha^2}\mu\right]^{-1/2}\), and \(D = \sigma\sqrt{\mu}\) with the variable transformation \(y = u^{-\sigma}\)?

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