Given that

2t² y′′ + ty′ - 3 y = 0

Show that this given solution are form a fundamental set of solutions for the differential equation?

Solution

The two solutions from that illustration are

y₁(t) = t^-1          y₂(t) = t^3/2

Let's calculate the Wronskian of these two solutions.

Therefore, the Wronskian will never be zero. Memorizes that we can't plug t = 0 in the Wronskian. It would be a problem in determining the constants in the general solution, except that we as well can't plug t = 0 in the solution either and thus this isn't the problem which it might appear to be.

So, as the Wronskian isn't zero for any t the two solutions form a fundamental set of solutions and the general solution is as

y(t) = c₁t^-1+ c₂ t^3/2as we claimed in that illustration.

To this point we're determined a set of solutions then we've claimed which they are actually a fundamental set of solutions. Certainly, you can now verify all those claims which we've made, though this does bring up a question.