math121a-f23:october_23_monday
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math121a-f23:october_23_monday [2023/10/23 05:16] pzhou created |
math121a-f23:october_23_monday [2026/02/21 14:41] (current) |
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| ====== Oct 23: constant coeff diffrential equation ====== | ====== Oct 23: constant coeff diffrential equation ====== | ||
| - | warm-up: how to solve the equation | + | ====warm-up: |
| + | how to solve the equation | ||
| $$ (z-a) (z-b) = 0 $$ | $$ (z-a) (z-b) = 0 $$ | ||
| well, we would have two solutions $z=a$ and $z=b$ (assuming $a \neq b$). | well, we would have two solutions $z=a$ and $z=b$ (assuming $a \neq b$). | ||
| + | ==== 1 ==== | ||
| how to solve equation ( $a \neq b$) | how to solve equation ( $a \neq b$) | ||
| $$ (d/dx - a) (d/dx - b) f(x) = 0? $$ | $$ (d/dx - a) (d/dx - b) f(x) = 0? $$ | ||
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| wait, how do you know you have found ALL the solutions? how do you know you didn't miss any? There is a theorem saying that, a constant coeff ODE of order $n$ will have a $n$-dimensional solution space. | wait, how do you know you have found ALL the solutions? how do you know you didn't miss any? There is a theorem saying that, a constant coeff ODE of order $n$ will have a $n$-dimensional solution space. | ||
| + | ==== 2 ==== | ||
| How about the case where $a=b$? | How about the case where $a=b$? | ||
| $$ (d/dx - a) (d/dx - a) f(x) = 0? $$ | $$ (d/dx - a) (d/dx - a) f(x) = 0? $$ | ||
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| That means $g(x) = c_0 + c_1 x$. Hence we get the claimed general solution | That means $g(x) = c_0 + c_1 x$. Hence we get the claimed general solution | ||
| + | ==== 3 ==== | ||
| + | Initial condition / Boundary condition. | ||
math121a-f23/october_23_monday.1698038166.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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