Reduction of order - fundamental set of solutions, Mathematics

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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) = t3/2

Let's calculate the Wronskian of these two solutions.

1711_Reduction of Order - fundamental set of solutions.png

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 t3/2as 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.


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