Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.


Having some trouble. Im not sure what d_t(u) and :u(with the . above it) means. Would appreciate some help. Thanks

share|improve this question
The "overdot" notation for the derivative with respect to time is due to Newton. The notation now has limited popularity. –  André Nicolas Oct 8 '11 at 22:39

2 Answers 2

The first several things listed are just different notations for the derivative of u with respect to t: $du/dt$, $d_t(u)$, and $\dot{u}$ are all the same thing.

You're asked to solve $u'-g(t)u=h(t)$ with initial condition $u(0)=u_0$.

The trick to solving such an equation is to multiply by some function so that the left hand side is the result of differentiating using the product rule: Say multiply by $X$ and get $Xu'-Xg(t)u = Xh(t)$. We want...

$$Xu'+X'u=\frac{d}{dt}\left[Xu\right] = Xu'-Xg(t)u$$

Thus we need $X'u=-Xg(t)u$ that is $X'=-Xg(t)$ so that $X=exp(-\int_0^t g(x)\,dx)$

Hopefully this will get you started. By the way this kind of equation is a "first order linear ODE". http://en.wikipedia.org/wiki/Linear_differential_equation (see first order)

share|improve this answer

$\mathrm d_t u$ and $\dot u$ are both just alternative notations for $\frac{\mathrm du}{\mathrm dt}$.

In fact, that is what the $=:$ symbol means: "The symbol to the right of $=:$ is is hereby defined to be a name for the thing to the left".

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.