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Show that the solution set $S$ of the second-order differential equation $y^{\prime \prime}+a y^{\prime}+b y=0$ is a subspace of $\mathscr{F}$...

Question

Show that the solution set $S$ of the second-order differential equation $y^{\prime \prime}+a y^{\prime}+b y=0$ is a subspace of $\mathscr{F}$

Show that the solution set $S$ of the second-order differential equation $y^{\prime \prime}+a y^{\prime}+b y=0$ is a subspace of $\mathscr{F}$



Answers

Prove that if $y_{1}$ and $y_{2}$ are linearly independent solutions
of $y^{\prime \prime}+p y^{\prime}+q y=0$ on $(a, b),$ then they cannot both
be zero at the same point $t_{0}$ in $(a, b) .$


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