I am trying to solve a time dependent second order differential equation with a known solution. The equations I want to solve are

$ x & # 39; & # 39; (t) = frac {-x (t)} {(x (t) ^ 2 + y (t) ^ 2) ^ {1.5}} $

and

$ y & # 39; & # 39; (t) = frac {-y (t)} {(x (t) ^ 2 + y (t) ^ 2) ^ {1.5}} $

This clearly has the solution x (t) = Cos (t + a), y (t) = Sin (t + b), where a and b can be defined according to the boundary conditions. However, when I try to solve this problem in Mathematica, it is impossible to return the solution, even if I include the boundary conditions.

```
eqns1 = {(x(t)^2 + y(t)^2)^-1.5*x''(t) == -
x(t), (x(t)^2 + y(t)^2)^-1.5*y''(t) == - y(t), x(0) == 0,
y(0) == 1};
DSolve(eqns1, {x, y}, t)
```

But it comes back

```
DSolve({(x^(Prime)(Prime))(t)/(x(t)^2 + y(t)^2)^1.5 == -x(t), (
y^(Prime)(Prime))(t)/(x(t)^2 + y(t)^2)^1.5 == -y(t), x(0) == 0,
y(0) == 1}, {x, y}, t)
```

that is, it just tells me that Mathematica cannot solve this problem. Does anyone know why i can't find a solution or what is the problem? Is there a known way around this problem?

Thank you