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

 

[blue_raver22]

I would say that solving second order ordinary differential equations (ODEs) is an important task in many fields of science, including physics, engineering, and mathematics. In this particular case, we are dealing with an ODE that involves the first and second derivatives of a function u with respect to a variable s, as well as a constant k.

The equation can be rewritten in a more standard form as d^2u/ds^2 + k^2 = cos(u)(du/ds)^2. This type of ODE is known as a second order nonlinear ODE, as it involves both the function u and its derivatives. In general, solving nonlinear ODEs is more challenging than solving linear ODEs, but there are various techniques and methods that can be used to find solutions.

One approach to solving this ODE could be to use a numerical method, such as the Runge-Kutta method, to approximate the solution. This method involves breaking the problem into smaller steps and using iterative calculations to find an approximate solution. Another approach could be to use an analytical method, such as separation of variables or the method of undetermined coefficients, to find a closed-form solution. However, these methods may not always be possible or practical, depending on the complexity of the equation.

In conclusion, solving this second order nonlinear ODE requires careful analysis and the use of appropriate techniques. The solution will depend on the specific values of k and the initial conditions of the problem. As a scientist, it is important to carefully consider the assumptions and limitations of any solution method used and to verify the results through experimentation or comparison with other known solutions.