Differential equation: capacitor

calculus ordinary-differential-equations

2,546

Solution 1

$$\frac {du} {dt} + \frac {1} {RC} u = 0$$ $$\frac {du} {dt} =- \frac {u} {RC} $$ $$\frac {du} {u} =- \frac {dt} {RC} $$ Integrate $$\int \frac {du} {u} =-\int \frac {dt} {RC} $$ $$\ln(u)=- \frac {t} {RC} +K $$ $$u(t)=Ke^{- \frac {t} {RC}} $$

Solution 2

Hint:This is equivalent to $\frac{(\frac{du}{dt})}{u}=-1/RC$. Now LHS is just $\frac{d \log u}{dt}$.

Solution 3

Assuming $R$ and $C$ are constant this has the solution

$$u(t) = e^{-t\frac{1}{RC}}$$

This is a first-order homogeneous ODE, for which general solutions are easily availble. Khan Academy has a good tutorial, for example.

2,546

gbgult

Updated on July 23, 2022

Comments

gbgult over 1 year

The voltage of a capacitor can be described with the differential equation $ \frac {du} {dt} + \frac {1} {RC} u = 0$ where the voltage is u(t) at the time t.

Solve the differential equation:

Don't really know how to solve this one. Would appreciate tips/hints on how to tackle differential equations like this in general.
- Paul over 5 years
  
  This is the most basic first order homogeneous constant coefficient differential equation. Look in any engineering maths textbook.
- snulty over 5 years
  
  A usual idea for a homogenous equation with constant coefficients is to try an exponential solution $u=e^{\lambda t}$. On the other hand since it's first order (one derivative) you can try separate the variables, bring $u$'s to one side $t$'s to the other
- gbgult over 5 years
  
  Yeah, I misread the question and didn't realise that R and C were constants. Thus my confusion. I'll leave the question be though because someone might find it useful.
- hardmath over 5 years
  
  If you want to leave it, that's fine by me. It is worth noting that this example exhibits exponential decay because $\lambda = -1/RC$ is negative. Whatever (voltage) charge is initial placed on the capacitor, it will be discharged more quickly with lower resistance $R$.