$Dependency of a solution for differential equation$

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$Dependency of a solution for differential equation$

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. $

Ordinary Differential Equations

Danielatia123

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Since any function in $V_j$ is a polynomial times $e^{r_j x}$, a linear dependence relation for $y_j$'s means $$ p_1e^{r_1 x}+p_2e^{r_2 x}+p_3e^{r_3 x}=0$$ for some polynomials $p_j$ of degree at most two. But then we have $$ p_1=-p_2e^{(r_2-r_1) x}-p_3e^{(r_3-r_1) x}$$ Now the 3rd derivative of $p_1$ is zero but no derivative of the right hand side is zero unless the whole right hand side is zero. This means that $p_1=0$ and $$ p_2e^{(r_2-r_1) x}+p_3e^{(r_3-r_1) x}=0\implies p_2=p_3e^{(r_3-r_2) x}$$ Again the 3rd derivative of $p_2$ is zero but no derivative of the right hand side is zero unless the whole right hand side is zero. Hence we get $p_2=0$ and $$p_3e^{(r_3-r_2) x}=0\implies p_3=0$$ So any linear dependence relation between $y_j$'s must be trivial and therefore they are linearly independent.

Martin

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Danielatia123

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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.
Danielatia123

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Nevermind. My mistake
Danielatia123

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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
Martin

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The 3rd derivative is 0 for 2nd order polynomials, but 4th order derivative works as well.
Danielatia123

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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?
- Martin
  
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  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.

$Dependency of a solution for differential equation$

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