# Dependency of a solution for differential equation

Let $Let \ r_{1}, \ r_{2},\ r_{3}\in R \ distinct.\\ Define: V_{j}=span\left\{ { \ {e ^ {r_{j}x}} , xe ^ {r_{j}x} , x^2e ^ {r_{j}x}} \right\} ,j=1,2,3,. \\ Show \ that \ for \ every \ 3 \ functions \ y_{1}\in V_{1} , y_{2}\in V_{2} , y_{3}\in V_{3} \ which \ non \ of \ them \ are \ the \ zero \ function \ , \ y_{1},y_{2},y_{3} \ are \ linearly \ independent \ in \ every \ interval.$

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• I think you have a mistake. The second derivative of p1,p2,p3 should be zero and not the third one.

• Nevermind. My mistake

• But i think that the 4th derivative should be zero , because linear combination of Vj is a solution of differential equation with constant co with characteristic polynomial with 3 algebraic multiplicity

• The 3rd derivative is 0 for 2nd order polynomials, but 4th order derivative works as well.

• Just a second. I think I got confused. if with characteristic polynomial with 3 algebraic multiplicity then it means that the deg of the poly is 3 and it means that the 4 derivative would be zero , no?

• I am not sure what you are referring to. But here the polynomials are at most 2nd degree by definition of V_j. So the 3rd derivative is 0. The 4th derivative and higher derivatives are also 0 of course.