Solution
= Solution
For $k\ne m$, the required Euler-equation solution is
$$
x(t)=\frac{t^k-t^m}{k^2-m^2}.
$$
For $k=m>0$, the limiting resonant form is
$$
x(t)=\frac{t^m\log t}{2m}.
$$
Both extend continuously to zero and satisfy the two <boundary conditions>. If $k=m=0$, integration gives
$$
x=\frac12(\log t)^2+C\log t+D,
$$
which cannot have a finite zero <limit> at $t=0$, so no such solution exists.