From 81b0363ce58c250fdd31c72911bb4e16a077b3d0 Mon Sep 17 00:00:00 2001 From: Sasserisop Date: Tue, 10 Oct 2023 17:52:23 -0600 Subject: [PATCH] revise mouc and cauchy --- content/Cauchy-Euler equations (lec 10).md | 2 +- content/Method of undetermined coefficients (lec 8-9).md | 5 +++-- 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/content/Cauchy-Euler equations (lec 10).md b/content/Cauchy-Euler equations (lec 10).md index 1776995..2dc621e 100644 --- a/content/Cauchy-Euler equations (lec 10).md +++ b/content/Cauchy-Euler equations (lec 10).md @@ -36,7 +36,7 @@ $A=\frac{1}{2}$ general solution in terms of t: $y(t)=c_{1}e^{-t}+c_{2}te^{-t}+\frac{1}{2}t^2e^{-t}$ but we want solution in terms of x: -$y(x)=c_{1}e^{-\ln(x)}+c_{2}\ln(x)e^{-\ln(x)}+\frac{1}{2}\ln(x)^2e^{-\ln(x)}$ +$y(x)=c_{1}e^{-\ln(x)}+c_{2}\ln(x)e^{-\ln(x)}+\frac{1}{2}\ln(x)^2e^{-\ln(x)}$ <- This is rather lousy notation, the y here isn't the same as the y above. Conceptually though, it's all oke doke. $$y(x)=c_{1}x^{-1}+c_{2}\ln(x)x^{-1}+\frac{1}{2}{\ln(x)^2}x^{-1}$$ We are done. diff --git a/content/Method of undetermined coefficients (lec 8-9).md b/content/Method of undetermined coefficients (lec 8-9).md index fa6a019..c188499 100644 --- a/content/Method of undetermined coefficients (lec 8-9).md +++ b/content/Method of undetermined coefficients (lec 8-9).md @@ -102,8 +102,9 @@ s=0, if r is not a root, s=1 if r is a single root, s=2 if r is a double root. -case ii) $ay''+by'+cy=P_{m}(t)e^{\alpha t}\cos(\beta t)+P_{m}(t)e^{\alpha t}\sin(\beta t)$ -Then we guess the particular solution is of the form: $y_{p}(t)=t^s[(A_{k}t^k+A_{K-1}t^{k-1}+\dots+A_{0})e^{\alpha t}\cos(\beta t)+(B_{k}t^k+B_{k-1}t^{k-1}+\dots+B_{0})e^{\alpha t}\sin(\beta t)]$ +case ii) $ay''+by'+cy=P_{m}(t)e^{\alpha t}\cos(\beta t)+Q_{n}(t)e^{\alpha t}\sin(\beta t)$ +Then we guess the particular solution is of the form: $y_{p}(t)=t^s[(A_{k}t^k+A_{k-1}t^{k-1}+\dots+A_{0})e^{\alpha t}\cos(\beta t)+(B_{k}t^k+B_{k-1}t^{k-1}+\dots+B_{0})e^{\alpha t}\sin(\beta t)]$ where: +k=max(m,n) s=0 if $\alpha+i\beta$ is not a root, s=1 if $\alpha+i\beta$ is a root. \ No newline at end of file