Solving 2nd Order ODE: du/ds & k^2

[lavinia]

d^{2}u/ds^{2}= cosu[(du/ds)^{2} - k^{2}]

 

[HallsofIvy]

"cosu" is cos(u)? Since the independent variable, s, does not appear explicitely in that equation you can use a technique called "quadrature".

Let v= du/ds so that d^2u/ds^2=dv/ds but, by the chain rule, dv/ds= (dv/du)(du/ds)= v(dv/du) so your equation becomes v dv/du= cos(u)(v^2+ k^2)

That's a separable first order equation. Once you have solved it for v, integrate to find u.

 

[jackmell]

d^{2}u/ds^{2}= cosu[(du/ds)^{2} - k^{2}]

That lends itself to interpretation:

\frac{d^2 u}{ds^2}=\cos(u(v))\biggr|_{v=(u')^2-k^2}

that's doable right?

 

[lavinia]

"cosu" is cos(u)? Since the independent variable, s, does not appear explicitely in that equation you can use a technique called "quadrature".

Let v= du/ds so that d^2u/ds^2=dv/ds but, by the chain rule, dv/ds= (dv/du)(du/ds)= v(dv/du) so your equation becomes v dv/du= cos(u)(v^2+ k^2)

That's a separable first order equation. Once you have solved it for v, integrate to find u.

thanks - pretty cool