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Matrix differential equation $\dfrac{dx}{dt} = x^t$ as a ODE

What is the general solution to $\dfrac{dx}{dt} = x^t$?


Hint: Substitute $y=x^t$ to get $\frac{dy}{dt}=y$ and then use a standard integrating factor or a solution of the form $f(t)y(t)$.

Added: The latter is actually a general solution, since the ODE is linear and homogeneous. This is especially easy when $t$ is real. We have $x(t)=k\,e^{\int\! a(t)\,dt}$, where $k$ is a constant and $a(t)$ is the time function. The integral can be taken back in the original differential equation: we get $d\left(\frac{1}{k\,e^{\int\! a(t)\,dt}}\right)=a(t)\,dt$. Use the given solution in this case: $a(t)=t$ and we get $x(t)=k\,e^{\int\! t\,dt}=kt$ as required. If $t$ is real we get $\lim\limits_{t\to0}\frac{1}{t}=\lim\limits_{t\to0}\frac{1}{kt}=0$ and thus the solution is $\lim\limits_{t\to0}\

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