I have an ODE for a vector $\vec{w} \in \mathbb{R}^n$. $$ \frac{\text{d}}{\text{d}t} \vec{w}(t) + \frac{\alpha}{n} XX^T \vec{w}(t) = \frac{\alpha}{n} X \vec{y} $$ where $X \in \mathbb{R}^{n\times m}$, $\alpha > 0$ is a constant and $y\in \mathbb{R}^m$ is also constant. I'm having a hard time constructing an analytical solution from an integration constant. We see that the integration constant is $$ I(t) = \exp \left( -\int \frac{\alpha }{n} XX^T \text{d}t\right) = \vec{w_0} \exp \left(-\frac{\alpha t}{n} XX^T \right). $$ I know that the solution is $$ \vec{w}(t) = \exp \left(-\frac{\alpha t}{n} XX^T \right) \vec{w_0} + \left(I_n - \exp \left(-\frac{\alpha t}{n} XX^T \right) \right)\left(XX^T\right)^{-1}X\vec{y} $$ but I get nowhere close to that so I assume my integration constant is false. Perhaps there's another formula for matrix valued function's since I've assume it's the same as a classic first order inhomogenous linear ode.
EDIT: We known that $n < m$.