query: If the characteristic equation of a second ODE has a double root, why do we have to introduce $xe^{rx}$?
positive_passages.docid: 11654
positive_passages.text: Where does $xe^x$ solution come from when the characteristic polynomial is square?
negative_passages.docid: 61477
negative_passages.text: How do I resolve a recurrence relation when the characteristic equation has fewer roots than terms?
negative_passages.docid: 27812
negative_passages.text: Understanding the differential $dx$ when doing $u$-substitution
negative_passages.docid: 60396
negative_passages.text: How to Solve the Coupled Differential Equations?
negative_passages.docid: 153580
negative_passages.text: Solutions of autonomous ODEs are monotonic
negative_passages.docid: 30723
negative_passages.text: Second derivative "formula derivation"
negative_passages.docid: 38839
negative_passages.text: Exponential of the differential operator
negative_passages.docid: 145504
negative_passages.text: Help solving variable separable ODE: $y' = \frac{1}{2} a y^2 + b y - 1$ with $y(0)=0$
negative_passages.docid: 8145
negative_passages.text: Comparison theorem for ODE
negative_passages.docid: 127582
negative_passages.text: Solving of a seperable differential equation
negative_passages.docid: 58083
negative_passages.text: what can I say about the solution $y(x)$ of the ODE?
negative_passages.docid: 93590
negative_passages.text: system of ode with non-constant coefficient matrix
negative_passages.docid: 67641
negative_passages.text: Why should field operators satisfy the classical equations of motion?
negative_passages.docid: 45811
negative_passages.text: How to prove that r is a double root if and only if it is a root of a polynomial and of its derivative.
negative_passages.docid: 132239
negative_passages.text: Why is the solution single-valued?
negative_passages.docid: 118072
negative_passages.text: Division of differential equations
negative_passages.docid: 131786
negative_passages.text: Why are second order linear PDEs classified as either elliptic, hyperbolic or parabolic?