## Find the solution of the linear differential equation of second order

**Differential equations** Using the slogan "The difference of any two solutions of the equation are not homogeneous, it is a solution of the homogeneous", is the solution of the linear differential equation of second order, knowing that the three solutions of the non-homogeneous associated, are: 
\begin{align}&ϕ1(τ)=t2,ϕ2(τ)=t2+e2t,ϕ3(τ)=1+t2+2e2t \end{align}how Many functions are linearly independent, you need to find to give the general solution of a homogeneous linear differential equation of order 4? Does this contradicts the theorem of Existence and Uniqueness? Justified.