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How can I show that $y_1(t)$ and $y_2(t)$ form a fundamental set of solutions? I know that a fundamental set of solutions can be defined as $y(t)=c_1y_1(t)+c_2y_2(t)$

More specifically, I need to show that the following form a fundamental set of solutions:

a. $y_1(t)=\sqrt t$ and $y_2(t)=1/t$

$2t^2y"+3ty'-y=0$

So I decided to calculate the Wronskian and got: $W[y_1,y_2](t)=\frac{-3}{2 \sqrt{t^3}}$ which can never be equal to 0, so I think this means the solutions are linearly independent.

$y(t)=c_1y_1(t)+c_2y_2(t)$ $=c_1\sqrt t+c_2(1/t)$

And then I am kind of stuck..

b. $y_1(t)=e^{-t^2/2}$ and $y_2(t)=e^{-t^2/2}\int_0^te^{s^2/2}ds$

$y"+ty'+y=0$

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Any two linearly independent solutions of a homogeneous second-order linear DE form a fundamental set. Once you know that the Wronskian is nonzero, all you have to do is verify that $y_1$ and $y_2$ actually are solutions (which you do by plugging them in to the differential equation).

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