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Reduction to Riccati by giving the symmetry to dsolve
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The reduced ODE above is of Riccati type:
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Converting this ODE into a first order ODE of Riccati type
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In the answer returned by convert, there are the Riccati ODE and the transformation of variables used. Changes of variables in ODEs can be performed using ?PDEtools[dchange]. For example, using the transformation of variables above, we can recover the result returned by convert.