# Thread: first order differential equation from variational calculus problem has me stumped!

1. ## first order differential equation from variational calculus problem has me stumped!

This arises when solving the problem of Dido,

$\displaystyle \frac{\dot x}{\sqrt{1+\dot x^2}} = -\frac{t}{\lambda} + C$, where C and $\displaystyle \lambda$ constant

They mention that it can be solved by direct integration or by using the substitution $\displaystyle \dot x =tan(\alpha)$.

The solution is given as circles of the form $\displaystyle (\frac{t}{\lambda}-d)^2+(\frac{t}{\lambda}-k)^2 = 1$ where d and k are constants of integration.

I don't know how to deal with that many differentials of x on the same side of the equation! I need some guidance here !

2. Well, what do you get on the LHS when you do the indicated substitution?

3. i get sin ? Let me work on it a little more, thanx for the tip !

4. Sounds good so far. Keep going!

5. First of all substitute z for (C - t/λ) on the R.H.S. and don't forget to change from dx/dt
to dx/dz on the L.H.S. Then solve for dx/dz , to get it as a function of z.Now integrate both sides using tables....