At t=0 my equation suggests that v=5v. This is not true as it should be 0 because capacitors cannot change their voltage instantaneous.In addition to the abysmal failure to properly track units (which is almost certainly because you were never expected to do so), or even bother to tack units onto the final result (which you probably are at least expected to do), you failed to ask if the answer makes sense.
At t = 0 s, what is the voltage across the capacitor?
What does your result say the voltage across the capacitor is at t = 0 s.
The only educational value to be derived from transient analysis by differential equation is to convince the student that first-order RC or RL circuits excited by step functions (ON-OFF switches), which are the most common situation encountered, always exhibit an exponential rise or exponential decay response that can be completely characterized by an initial value, final value, and time constant.I suspect, at this point in many first circuits courses, that they are just starting with transient response in first order circuits,
Correct. These are the kinds of questions you want to get in the routine habit of asking yourself both as you are solving problems and once you get to the final result.At t=0 my equation suggests that v=5v. This is not true as it should be 0 because capacitors cannot change their voltage instantaneous.
This is a very dangerous approach -- and I suspect it is how you got the wrong equation in the first place. By "just need to add to the equation" you are practicing the Art of the Happening, which comes down to throwing things around more-or-less randomly and hoping that, eventually, you'll stumble across something that happens to work.So would I just need to add to the equation to make v=0v when t=0?
\(v(t) = 5(1-e^{-100t})\) This equation makes it so that for t=0 the result would be 5(1-1) = 0v.
And this error didn't get caught because you didn't track your units. If you had, you would have seen that your exponent ended up with units of time-squared, and since the exponent must be dimensionless, you would have known that the answer had to be wrong.Edit: Tau in OP should be 100 also.
Your point?The only educational value to be derived from transient analysis by differential equation is to convince the student that first-order RC or RL circuits excited by step functions (ON-OFF switches), which are the most common situation encountered, always exhibit an exponential rise or exponential decay response that can be completely characterized by an initial value, final value, and time constant.