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more fixes for voparam

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Sasserisop 2023-10-08 00:26:07 -06:00
parent 1761284b70
commit bbfe8f4123
1 changed files with 2 additions and 2 deletions
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content/Variation of parameters (lec 9-10).md
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@ -11,8 +11,8 @@ $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ <- btw $y_{1}$ and $y_{2}$ are ofte
Impose the following: Impose the following:
1) $v_{1}'y_{1}+v_{2}'y_{2}=0$ 1) $v_{1}'y_{1}+v_{2}'y_{2}=0$
Compute the derivatives and simplify: Compute the derivatives and simplify:
$y'_{p}=v_{1}y_{1}'+v_{2}y_{2}'$ $y_{p}''=v_{1}y_{1}'+v_{2}y_{2}'$
$y_{p}''=v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}''$ $y_{p}'=v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}''$
Now we plug those into the second order equation and simplify: Now we plug those into the second order equation and simplify:
2) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$ 2) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$
We now have a system of two equations (1 and 2). Now we can solve for $v_{1}$ and $v_{2}$: We now have a system of two equations (1 and 2). Now we can solve for $v_{1}$ and $v_{2}$: