I want to find the general solution to the system $ty' = Ay + b(t)$ where $A$ is a constant matrix and $b(t)$ is a continuous function.
I know that given a system in the form $y' = Ay + b(t)$, the particular solution is written as $y_p = Y(t)\int Y^{-1}(t)b(t)dt$ where $Y(t)$ is a matrix whose ith column is the ith linearly independent solution to the system $y' = Ay$. Then the general solution to $y' = Ay + b(t)$ is written as $$y(t) = y_c(t) + y_p(t)$$
For $ty'(t) = Ay(t) + b(t)$, I am not sure how dividing by $t$ on both sides affects the particular solution if at all. If I divide by $t$, I get $y'(t) = Ay(t)\frac{1}{t} + b(t)\frac{1}{t}$, could I simply relabel $y(t)$ and $b(t)$ to include the $\frac{1}{t}$ term?
Best Answer
Follow one of the strategies for Euler-Cauchy equations. Set $t=e^s$, $u(s)=y(e^s)$, then $$ u'(s)=e^sy'(e^s)=Ay(e^s)+b(e^s)=Au(s)+b(e^s) $$ which is now a usual linear system with constant matrix $A$.