Question

Do the functions `y_1(t)=√t` and `y_2(t)=1/t` form a fundamental set of solutions of the equation `2t^(2)y''+3ty'-y=0`,on interval `0<t<∞` ? Justify your answer.

Answer 1

See below.

Explanation:

The functions `y_1(t), y_2(t)` have similar structure so substituting into the differential equation `y(t) = t^{\alpha}` we have

# t^{alpha}(2alpha^2+alpha-1) = 0 #

and this is true for `t \ne 0` when `alpha = -1, alpha=\frac{ 1}{2}` corresponding to `y_1` and `y_2` hence `y_1` and `y_2` are a fundamental set of solutions