Can Nonlinear Boundary Value Problems Be Solved Analytically?

[amr07]

Dear everybody,

I am new here. I need your suggesstion and helping for solving the next nonlinear boundary value problem:

x'=-y/sqrt(x^2+y^2)
y'=x/sqrt(x^2+y^2)
with x(0)=y(0)=-1 and x(pi/4)=y(pi/4)=1

so is it easy to get the analytical solution to the problem above or I have to solve this problem numerically.
thanks in advance

Amr

 

[elibj123]

Using polar coordinates, and rewriting in vector form you get the equivalent equation:

$$\frac{d\vec{r}}{dt}=\hat{\theta}$$ (1)

$$\frac{d\vec{r}}{dt}=\hat{r}\frac{dr}{dt}+r\frac{d\theta}{dt}\hat{\theta}$$

Putting this in the equation (1) and comparing different components you get:

$$\frac{dr}{dt}=0 => r(t) \equiv const=R$$
$$r\frac{d\theta}{dt}=1$$

Since r is constant you get:

$$\theta=\frac{t}{r}+\varphi$$

From this you get that:

$$x(t)=Rcos(\frac{t}{R}+\varphi); y(t)=Rsin(\frac{t}{R}+\varphi);$$

You have a circular motion with a radius and initial phase to determine from boundary conditions.