Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What are the solutions to the diff eqn: $A\dot{x} + \cos(x) - 1 = B$ subject to boundary conditions $\lim\limits_{x \to -\infty} x(t) = 0$ and $\lim\limits_{x \to +\infty} x(t) = C$, where $ A, B, C$ are constants?

share|cite|improve this question
Try writing $\dot{x} = dx/dt$, separate them to two sides, like $\displaystyle A\frac{dx}{1-\cos(x)} = B\,dt$, and integrate on both sides. – Shuhao Cao Aug 11 '11 at 18:35
@MathChief Shouldn't it be $\displaystyle A\frac{dx}{B+1-\cos(x)} = dt$ – Michael Banaszek Aug 11 '11 at 18:45
Are these limits when $t$ (and not $x$) goes to $\pm\infty$? And are the signs of $A$, $B$ and $C$ known? – Did Aug 11 '11 at 18:46
@Michael, that's right, my bad. – Shuhao Cao Aug 11 '11 at 19:08

Separation of variables gives $ dx/(p + q\cos x) = dt$ for some $p$ and $q$ and this can be integrated on any (maximal) interval for which the left side is defined, to give $F(x)=t$ for an explicit function $F$. On such an interval $F$ will be monotonic and with range from $-\infty$ to $+\infty$; the solutions of the differential equation are $x(t) = F^{-1}(t+K)$ for constant $K$. Boundary behavior at infinity should be $x(\pm \infty) = \pm \infty$ according to the sign of $p+q\cos x$ in the interval.

If these calculations are correct then there is no freedom to define the boundary conditions at infinity, only at finite times.

share|cite|improve this answer
In the expression $x(\pm\infty)=\pm\infty$ the first $\pm\infty$ should be replaced by the bounds of the maximal interval of integration of the differential equation. – Did Aug 11 '11 at 22:24
@Didier: yes, thanks. What I should have written is that the boundary behavior is $F(\pm \infty) = \pm \infty$ which is, except for the sign, independent of $A, B$ and $C$. Describing this in terms of $x$ (i.e., the maximal interval) does utilize the constants in the problem statement. – zyx Aug 12 '11 at 0:15
thank you, zyx! – quirkyquark Aug 13 '11 at 16:22

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.