homogeneous second order equation in D'

Question

please, if we consider in $\mathcal{D}'(\mathbb{R})$, the equation of second order $$ T"+ aT' +b T=0 $$ where a and b are constants. How we pruve that the set of solutions of this equation is an vectoriel space of dimension 2?

Answer 1

The equation is linear. That is, if $T_1$ And $T_2$ are solutions and $c,d \in \Bbb R$, then $$(cT_1 + dT_2)'' + a(cT_1 + dT_2)' + b(cT_1 + dT_2)\\= cT_1'' + dT_2'' + acT_1' + adT_2' + bcT_1 + bdT_2\\=c(T_1'' + aT_1' + bT_1) + d(T_2'' + aT_2' + bT_2)\\= c0 + d 0 = 0$$ So $cT_1 + dT_2$ is also a solution. Thus the set of all solutions forms a vector space. Now, you just need to prove that it is of dimension $2$. Consider solutions $T_1$ with $T_1(0) = 0$ and $T_1'(0) = 1$ and $T_2$ with $T_2(0) = 1$ and $T_2'(0) = 0$.

Can you prove that all solutions are a linear combination of $T_1$ and $T_2$?