It's the one solution that satisfies the initial value condition y(0)=1. This idea is analogously applied to the case of systems of ODEs with constant coeffici

It's the one solution that satisfies the initial value condition y(0)=1. This idea is analogously applied to the case of systems of ODEs with constant coefficients. The system y'=y=Iy has a basis of solutions of the form (e^x 0 ... 0)^t, (0 e^x 0 ... 0)^t, etc. Written column-wise into a matrix, this gives us the Wronskian Y which satifies Y(0)=I, and this Y is exactly the matrix exponential e^(xI). More in general, the equation y'=Ay has the Wronskian Y=e^(xA) which satisfies Y(0)=1.