Taking 3 Math courses in the same quarter
my undergraduate institution required calc 1&2, LA, diff equ, multivariable calc IN THAT ORDER mainly due to the way the course material was set up or else you'd probably be struggling. They wanted you to have the "maturity" of having completed the freshman calc sequence before you took a proof oriented course in LA, then you needed stuff from LA to solve the coupled equations in the ODE course, and finally the multivariable calc class since it covered multiple differentiation and integration in about one 90 min lecture each then spent the rest of the semester on Green/Stokes/Divergence theorems, lots of PDE theory, and little bits of analysis. So you really needed to know your LA stuff for the vector calc portion and you needed to be solid with your ODEs for the stuff on fourier analysis and PDEs.
I'm pretty sure the ODE and multivariable were really heavy on systems of equations and on PDEs mainly because they were the last math classes required for physics majors and they wanted to make sure they had very solid backgrounds in power series, vector calculus, systems of ODEs and PDE theory so they could actually teach E&M + QM without worrying about spending lecture time on "remedial mathematics" or having to offer one of those math methods in physics courses taught through the undergraduate physics curriculum.
basically, see what each course covers and even if you don't know 100% of what you're looking at you might know if you can handle taking them all at the same time.
your ODE class may be set up where you do first order ODEs then 2nd order, then power series solutions and finally systems of ODEs at the very end. if that's the case, you'll probably have all the algebra you need to be fine with the eigenvalue/vector stuff.
if your multivariable class is set up where you spend lots of time on div/grad/curl and multiple integration, then get to vector calc at the end and don't touch on PDEs, then you'll be fine to not have had a full course on ODEs before taking your calc class ... same with LA, if vector calc isn't until the end, of the course, then you'll have had more than enough algebra to handle the calc class.
