As an exercise, try solving 
, for 
. You will see that when 
, all kinds of problems occur.
When the coefficient is large, we speak of a ``stiff'' equation (or system).
Good methods will converge when the timestep is small enough, but small enough may be
very small. To overcome this limitation, we will look at implicit methods.
The Implicit Euler is like a right-end-point methods for integration. Given the ODE y'=f(t,y), we solve
Of course the new value of y appears on the right, so we have a nonlinear equation to solve - say by Newton's method. Although we have extra work to do, implicit methods allow larger timesteps; often implicit schemes are stable with no limitation on the timestep (implicit Euler is one such).
Folks have come up with a fancy name, Quasi-implicit methods, for schemes that are implicit but terminate the nonlinear solve after one step of the Newton method. That is, consider
There is a linear solve to perform at this stage, but only once. Quasi-implicit methods usually allow larger timesteps than explicit methods, but usually not the arbitrary timestep of fully implicit methods.
One can get second order accuracy by averaging an implicit and explicit update:
Again, one faces a nonlinear equation (system) to solve. This so-called Crank-Nicholson method is stable without timestep restriction.