Can I follow a graduate course in PDE without having studied ODE Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based course on ODE but studied ODE's during my bachelor's degree in Electrical Engineering but I can't solve any fancy ODE'S except perhaps the famous  Riccati's equation (which I deal with all the time).Do I need to review any ODE or not?
 A: I would say yes in general. Of course this depends on what kind of course you take. The huge difference between ODEs and PDEs is that for ODEs one is usualy interested in finding a solution. For PDEs this proves to be way to complicated very fast. There are some simple cases where explicit solutions can be constructed, sometimes by reducing to an ODE. But most of the time in PDE the only thing that is possible to do is to show existence of a solution. For this having had a course in functional analysis should be a good starting points since at least for linear PDE (you will not look at nonlinear ones for a long time, they get truly hard) many of the techniques are from functional analysis.
Of course it depends a bit on the kind of course given. There are some that delight in actually constructing solutions, but as far as I see, in mathematics, most courses seem to regard this approach as interesting but somewhat futile, so they show you some simple examples for this and then move on to prooving existence and regularity for larger classes of problems. There is also the third approach to PDE, the numerical one, where you try to find numerical ways to approximate a solution, but this also goes more into the direction of functional analysis.
A: Short answer: no.
Long answer: noooooooooooooooooooooo.
Long serious answer:

PDEs are tough. Very tough. And there are two types of PDEs in general: The type we can generalize to ODEs and the type we cannot.
When solving PDEs, you consider your work finished when you reduced them to a set of ODEs, because solving the ODEs then should be "trivial". A typical example are first order quasilinear partial differential equations. They are annoying little beasts that take a lot of work to solve, mainly because they reduce to a system of five ordinary differential equations (if you have a PDE with two variables).
Solving ODEs is not simple at all, and solving a system of five of them is even worse, so unless you are very skilled in ODEs, you cannot solve even the simplest case of PDEs.
