From 8808993cd7c94a732d24822e1b98b9139f58dd9e Mon Sep 17 00:00:00 2001 From: Sasserisop Date: Sun, 8 Oct 2023 11:43:02 -0600 Subject: [PATCH] more voparam fixes --- .../Second order homogenous linear equations (lec 5-7).md | 3 ++- content/Variation of parameters (lec 9-10).md | 7 +++---- content/_index.md | 2 +- 3 files changed, 6 insertions(+), 6 deletions(-) diff --git a/content/Second order homogenous linear equations (lec 5-7).md b/content/Second order homogenous linear equations (lec 5-7).md index 6737b70..bac4992 100644 --- a/content/Second order homogenous linear equations (lec 5-7).md +++ b/content/Second order homogenous linear equations (lec 5-7).md @@ -195,6 +195,7 @@ let it be $y_{p}(t)$ then the sum of the solutions $y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{p}(t)$ must solve $ay''+by'+cy=f(t)$ -Theorem: If $a(t),\ b(t),\ c(t)$ are continuous on $I$ , then IVP: $a(t)y''+b(t)y'+c(t)y=f(t)$ ; $y(t_{o})=y_{o}$ \ , $y'(t_{o})=y_{1}$ has a unique solution. +Theorem: If $a(t),\ b(t),\ c(t)$ are continuous on $I$ , then IVP: $a(t)y''+b(t)y'+c(t)y=f(t)$ ; +where $y(t_{o})=y_{o}$ , $y'(t_{o})=y_{1}$ has a unique solution. we will do the proofs next class. #end of lecture 7 diff --git a/content/Variation of parameters (lec 9-10).md b/content/Variation of parameters (lec 9-10).md index 756cde6..a6c30cd 100644 --- a/content/Variation of parameters (lec 9-10).md +++ b/content/Variation of parameters (lec 9-10).md @@ -11,7 +11,7 @@ $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ <- btw $y_{1}$ and $y_{2}$ are ofte Impose the following: 1) $v_{1}'y_{1}+v_{2}'y_{2}=0$ Compute the derivatives and simplify: -$y'_{p}=v_{1}y_{1}'+v_{2}y_{2}'$ +$y_{p}'=v_{1}y_{1}'+v_{2}y_{2}'$ $y_{p}''=v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}''$ Now we plug those into the second order equation and simplify: 2) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$ @@ -22,6 +22,7 @@ also, $W[y_1,y_{2}]$ is the WroĊ„skian, and it equals to: $\det \begin{pmatrix}y Finally, the general solution is: $$y(t)=y_{h}+y_{p}\qquad \text{where}\qquad y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$$ ## What you need to remember +#remember So, what do you need to commit to memory? I believe memorizing these three is a good tradeoff between memory allocated and speed for when you're solving a #voparam problem: 1) $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ 2) $v_{1}'=-\frac{{f(t)y_{2}(t)}}{aW[y_{1},y_{2}]}$ @@ -74,9 +75,7 @@ $y(0)=0=c_{1}-y_{p}(0)=c_{1}-\frac{1}{5} \implies c_{1}=\frac{1}{5}$ $y'(0)=\frac{4}{5}=2c_{2}+v_{1}'(0)+2v_{2}(0)-\frac{1}{5} \implies c_{2}=1$ $$y(t)=\frac{1}{5}\cos(2t)+\sin(2t)-\frac{1}{5}e^t+v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$$ -#end of lecture 9 -#start of lecture 10 - +#end of lecture 9 #start of lecture 10 #ex #second_order #voparam #mouc $$y''-2y'+y=e^t\ln(t)+2\cos(t)$$ Find homogenous solution first: diff --git a/content/_index.md b/content/_index.md index aaea114..8f824a9 100644 --- a/content/_index.md +++ b/content/_index.md @@ -12,7 +12,7 @@ I have written these notes for myself, I thought it would be cool to share them. [Variation of parameters (lec 9-10)](variation-of-parameters-lec-9-10.html) [Cauchy-Euler equations (lec 10)](cauchy-euler-equations-lec-10.html) (raw notes, not reviewed or revised yet.) [Free vibrations (lec 11-12)](free-vibrations-lec-11-12.html) (raw notes, not reviewed or revised yet.) -[Resonance in free vibrations (lec 13-14)](resonance-in-free-vibrations-lec-13.html) +[Resonance & AM (lec 13-14)](resonance-am-13-14.html) [Laplace transform (lec 14)](laplace-transform-lec-14.html)