First-order LSEs with constant coefficients: Repeated roots (continued)

In an earlier blog post, I showed the solution to a first-order linear system of two (ordinary) differential equations with constant coefficients for the case where the characteristic equation of the (constant coefficient) square matrix has repeated roots. In particular, I considered the system \begin{array}{rcl}\frac{\mathrm{d}x}{\mathrm{d}t}&=&ax+by\\ \frac{\mathrm{d}y}{\mathrm{d}t}&=&cx+dy\\\end{array}, where a,b,c,d are real constants and x,y,t are real variables, for the case where  (a-d)^2+4bc=0 and a\ne d. Now I look at the case where a=d. Continue reading “First-order LSEs with constant coefficients: Repeated roots (continued)”

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First-order LSEs with constant coefficients: Repeated roots

I’m currently teaching an undergraduate course on ordinary differential equations using the 8th edition of Elementary Differential Equations by Rainville, Bedient, and Bedient (published in 1996 by Prentice Hall). I’ve always wanted to have a blog post containing some LaTeX (using the plug-in WP LaTeX), so in this blog post I’ll be showing my solution to one of the exercises in the book.

I will derive the solution to a first-order linear system of two (ordinary) differential equations with constant coefficients for the case where the characteristic equation of the (constant coefficient) square matrix has repeated roots.
Continue reading “First-order LSEs with constant coefficients: Repeated roots”