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I know how to solve equations (which does not involve constants) using the DSolve. However, I got an equation that involve constants, i.e. $dy/dt=(\Gamma \cos t+T)y-y^{3}$, where $\Gamma$ and $T$ are constants.

I manually solved it and able to find the solution. However, I want to verify my answer using Mathematica.

How to solve this equation using the DSolve command?

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DSolve usually is able to handle constants. Try

DSolve[y'[t] == (\[CapitalGamma] Cos[t]  + T) y[t] \[Minus] y[t]^3, y, t]
(*{{y -> Function[{t}, -(E^(1/2 (4 t + 2 \[CapitalGamma] Sin[t]))/Sqrt[C[1] - 2 Inactive[Integrate][-E^(2 (2 K[1] + \[CapitalGamma] Sin[K[1]])), {K[1], 1,t}]])]}, 
{y ->Function[{t},E^(1/2 (4 t + 2 \[CapitalGamma] Sin[t]))/Sqrt[C[1] - 2 Inactive[Integrate][-E^(2 (2 K[1] + \[CapitalGamma] Sin[K[1]])), {K[1], 1, t}]]]}}*)
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