English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the study of ordinary differential equations (ODEs) of the form$\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, tworelated phenomena can arise: resonance, where $f(x)\propto u(x)$ and$\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$for $n\geq 2$. We illustrate a method to generate exact solutions to theseproblems by taking a known homogeneous solution $u(x)$, introducing a parameter$\epsilon$ such that $u(x)\rightarrow u(x;\epsilon)$, and Taylor expanding$u(x;\epsilon)$ about $\epsilon = 0$. The coefficients of this expansion$\frac{\partial^k u}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ yield the desiredresonant or repeated root solutions to the ODE. This approach, whenever it canbe applied, is more insightful and less tedious than standard methods such asreduction of order or variation of parameters. While the ideas can beintroduced at the undergraduate level, we could not find any elementary oradvanced text that illustrates these ideas with appropriate generality.

Perturbing ordinary differential equations to generate resonant and repeated root solutions