Question

The solutions to the linear differential equation d2u/dt2 = u form a vector space (since combinations of solutions are still solutions). find two independent solutions, to give a basis for that solution space.

Answer 1

Serious high school. This is one of the few differential equations I can solve.

The usual particular solution is $u=e^t$ because $e^t$ is its own derivative.

An independent solution is $u=e^{-t}$ which has a negative sign in the first derivative which turns back to positive in the second.

The arbitrary linear combination spans the solution space:

$u= c_1 e^t + c_2 e^{-t}$

But we only are asked for the basis.

Answer:$\textrm{. } \quad e^t, \quad e^{-t}$