Radioactive decay series
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Question: Consider the following radioactive decay series,
where $25$% of $X_3$ decays to $X_4$ while the remaining $75$% of $X_3$ decays to $X_5$. $X_6$ is stable doesn't decay. Let $X_i(t)$ represent the amount of each substance $X_i$ at time $t$ where $i =1,2,..,6.$ Find the system of first order ODE's that model the evolution of $X_i(t)$.
My attempt:
Let $lambda_1,lambda_2,...,lambda_6$ be the decay constants for substances $X_1,X_2,...,X_6$ respectively. Then the evolution of $X_i(t)$ can be modelled by:
$displaystyle fracdx_1dt=-lambda_1x_1$
$displaystyle fracdx_2dt=lambda_1x_1-lambda_2x_2$
$displaystyle fracdx_3dt=lambda_2x_2-lambda_3x_3$
$displaystyle fracdx_4dt=0.25x_3-lambda_4x_4$
$displaystyle fracdx_5dt=0.75x_3-lambda_5x_5$
$displaystyle fracdx_6dt=lambda_4x_4+lambda_5x_5$
Not super familiar with these questions, is this correct? Thanks
calculus differential-equations physics
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Question: Consider the following radioactive decay series,
where $25$% of $X_3$ decays to $X_4$ while the remaining $75$% of $X_3$ decays to $X_5$. $X_6$ is stable doesn't decay. Let $X_i(t)$ represent the amount of each substance $X_i$ at time $t$ where $i =1,2,..,6.$ Find the system of first order ODE's that model the evolution of $X_i(t)$.
My attempt:
Let $lambda_1,lambda_2,...,lambda_6$ be the decay constants for substances $X_1,X_2,...,X_6$ respectively. Then the evolution of $X_i(t)$ can be modelled by:
$displaystyle fracdx_1dt=-lambda_1x_1$
$displaystyle fracdx_2dt=lambda_1x_1-lambda_2x_2$
$displaystyle fracdx_3dt=lambda_2x_2-lambda_3x_3$
$displaystyle fracdx_4dt=0.25x_3-lambda_4x_4$
$displaystyle fracdx_5dt=0.75x_3-lambda_5x_5$
$displaystyle fracdx_6dt=lambda_4x_4+lambda_5x_5$
Not super familiar with these questions, is this correct? Thanks
calculus differential-equations physics
2
You've missed out $lambda_3$ from 4th and 5th equations.
â user121049
Sep 1 at 10:47
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Question: Consider the following radioactive decay series,
where $25$% of $X_3$ decays to $X_4$ while the remaining $75$% of $X_3$ decays to $X_5$. $X_6$ is stable doesn't decay. Let $X_i(t)$ represent the amount of each substance $X_i$ at time $t$ where $i =1,2,..,6.$ Find the system of first order ODE's that model the evolution of $X_i(t)$.
My attempt:
Let $lambda_1,lambda_2,...,lambda_6$ be the decay constants for substances $X_1,X_2,...,X_6$ respectively. Then the evolution of $X_i(t)$ can be modelled by:
$displaystyle fracdx_1dt=-lambda_1x_1$
$displaystyle fracdx_2dt=lambda_1x_1-lambda_2x_2$
$displaystyle fracdx_3dt=lambda_2x_2-lambda_3x_3$
$displaystyle fracdx_4dt=0.25x_3-lambda_4x_4$
$displaystyle fracdx_5dt=0.75x_3-lambda_5x_5$
$displaystyle fracdx_6dt=lambda_4x_4+lambda_5x_5$
Not super familiar with these questions, is this correct? Thanks
calculus differential-equations physics
Question: Consider the following radioactive decay series,
where $25$% of $X_3$ decays to $X_4$ while the remaining $75$% of $X_3$ decays to $X_5$. $X_6$ is stable doesn't decay. Let $X_i(t)$ represent the amount of each substance $X_i$ at time $t$ where $i =1,2,..,6.$ Find the system of first order ODE's that model the evolution of $X_i(t)$.
My attempt:
Let $lambda_1,lambda_2,...,lambda_6$ be the decay constants for substances $X_1,X_2,...,X_6$ respectively. Then the evolution of $X_i(t)$ can be modelled by:
$displaystyle fracdx_1dt=-lambda_1x_1$
$displaystyle fracdx_2dt=lambda_1x_1-lambda_2x_2$
$displaystyle fracdx_3dt=lambda_2x_2-lambda_3x_3$
$displaystyle fracdx_4dt=0.25x_3-lambda_4x_4$
$displaystyle fracdx_5dt=0.75x_3-lambda_5x_5$
$displaystyle fracdx_6dt=lambda_4x_4+lambda_5x_5$
Not super familiar with these questions, is this correct? Thanks
calculus differential-equations physics
calculus differential-equations physics
asked Sep 1 at 6:15
Sonjov
1056
1056
2
You've missed out $lambda_3$ from 4th and 5th equations.
â user121049
Sep 1 at 10:47
add a comment |Â
2
You've missed out $lambda_3$ from 4th and 5th equations.
â user121049
Sep 1 at 10:47
2
2
You've missed out $lambda_3$ from 4th and 5th equations.
â user121049
Sep 1 at 10:47
You've missed out $lambda_3$ from 4th and 5th equations.
â user121049
Sep 1 at 10:47
add a comment |Â
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2
You've missed out $lambda_3$ from 4th and 5th equations.
â user121049
Sep 1 at 10:47