Let $ \sum\limits_{k = 0}^n {y^{\left( k \right)} a_k } = 0 $ an homogeneous ODE, where $a_k$ are constants. How can I solve the equation when the roots are repeated? One way, that I saw in wikipedia, is using the fact that if $ e^{cx} $ is a solution, then $ (x^r)(e^{cx}) $ is also too, How can I prove this? It´s difficult to me, to evaluate the sum, because i want to show that $ \sum\limits_{k = 0}^n {\left( {x^r e^{cx} } \right)^{\left( k \right)} a_k } = 0 $ but I need to evaluate $ {\left( {x^r e^{cx} } \right)^{\left( k \right)} } $ and I don´t know how to do it Dx
linear ODE with constant coefficients, proof
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ordinary-differential-equations
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0The idea is to perturb coefficients of your equation a little so as to make all the roots distinct, and then consider the limit as the amount of perturbation goes to zero. – 2011-11-06