Help solving a Differential equation

k is an integer greater than 1.
$$y'=y^{\frac{-1}{k}}$$
solving this ode in matlab gives me this
$$(\frac{t*k+t+C1*k}{k})^{\frac{k}{1+k}}$$ which is correct since the further part of the problem proving a certain limit involving y follows from using the above matlab sol.

I am interested in solving this ode by hand.

by hand I am getting $$y^{\frac{k+1}{k}}=kt$$

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