Please analyze whether the following solution is correct.
We have the matrix differential equation about Q:
$$ \operatorname{tr} \left( \left( \textbf{Q+P} \right)^{-1}d\textbf{Q} \right) = \operatorname{tr}\left(\frac{\textbf{H}^T\textbf{H}d\textbf{Q}}{\left( r+\textbf{H}\left( \textbf{Q}+\textbf{P} \right)\textbf{H}^T \right)^2} \right) $$
where Q and P are symmetric positive definite matrices, P is independent of Q, r is a scalar constant, and H is a row matrix
$ \textbf{H}=\left[ 0, 0, ... 0, 1 \right] $.
$ tr\left[ \left( \textbf{Q+P} \right)^{-1}d\textbf{Q} \right]=tr\left[ \frac{\textbf{H}d\textbf{Q}\textbf{H}^T}{\left[ r+\textbf{H}\left( \textbf{Q}+\textbf{P} \right)\textbf{H}^T \right]^2} \right] $;
$ tr\left[ \left( \textbf{Q+P} \right)^{-1}d\textbf{Q} \right]=\frac{\textbf{H}d\textbf{Q}\textbf{H}^T}{n\left[ r+\textbf{H}\left( \textbf{Q}+\textbf{P} \right)\textbf{H}^T \right]^2} tr\left( \textbf{I} \right) $,
where I is the identity matrix and n is its rank.
We integrate both sides of the equation under the trace sign:
$ tr\left[\int_{}^{} \left( \textbf{Q+P} \right)^{-1}d\textbf{Q} \right]=tr\left[ \frac{\textbf{I}}{n} \right]\int_{}^{}\frac{\textbf{H}d\textbf{Q}\textbf{H}^T}{\left[ r+\textbf{H}\left( \textbf{Q}+\textbf{P} \right)\textbf{H}^T \right]^2} $.
We add the constants under the sign of the differential:
$ tr\left[\int_{}^{} \left( \textbf{Q+P} \right)^{-1}d\left( \textbf{Q+P} \right) \right]=tr\left( \frac{\textbf{I}}{n} \right)\int_{}^{}\frac{d\left(r+ \textbf{HQ}\textbf{H}^T +\textbf{HP}\textbf{H}^T\right)}{\left( r+\textbf{HQ}\textbf{H}^T+\textbf{H}\textbf{P}\textbf{H}^T \right)^2} $.
We integrate:
$ tr\left[ \ln\left( \textbf{Q+P} \right)+\textbf{C} \right]=-tr\left[ \frac{\textbf{I}}{n} \right]\left( \frac{1}{ r+\textbf{H}\left( \textbf{Q}+\textbf{P} \right)\textbf{H}^T}+c \right) $,
where C is a constant matrix, c is scalar constant . The above equation has an infinite number of solutions, but we can get one of all by getting rid of the matrix trace:
$ \ln\left( \textbf{Q+P} \right)=- \frac{1}{n\left[ r+\textbf{HQ}\left( \textbf{Q}+\textbf{P}\right)\textbf{H}^T \right]}\textbf{I}-\frac{c\textbf{I}}{n}-\textbf{C} $.
We get rid of the logarithm:
$ \textbf{Q+P} =\exp\left[ - \frac{1}{ n\left[ r+\textbf{H}\left( \textbf{Q}+\textbf{P}\right)\textbf{H}^T\right]} \right]\exp\left( -\frac{c\textbf{I}}{n}-\textbf{C} \right) $.
We obtain an implicit solution of the equation up to the constant matrix B:
$ \textbf{Q} =\textbf{B}\exp\left[ - \frac{1}{ n\left[ r+\textbf{H}\left( \textbf{Q}+\textbf{P}\right)\textbf{H}^T\right]} \right]-\textbf{P} $,
where
$ \textbf{B}=\exp\left( -\frac{c\textbf{I}}{n}-\textbf{C} \right) $.