Radioactive decay series

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Question: Consider the following radioactive decay series,



enter image description here



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










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  • 2




    You've missed out $lambda_3$ from 4th and 5th equations.
    – user121049
    Sep 1 at 10:47















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Question: Consider the following radioactive decay series,



enter image description here



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










share|cite|improve this question

















  • 2




    You've missed out $lambda_3$ from 4th and 5th equations.
    – user121049
    Sep 1 at 10:47













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Question: Consider the following radioactive decay series,



enter image description here



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










share|cite|improve this question













Question: Consider the following radioactive decay series,



enter image description here



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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asked Sep 1 at 6:15









Sonjov

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  • 2




    You've missed out $lambda_3$ from 4th and 5th equations.
    – user121049
    Sep 1 at 10:47













  • 2




    You've missed out $lambda_3$ from 4th and 5th equations.
    – user121049
    Sep 1 at 10:47








2




2




You've missed out $lambda_3$ from 4th and 5th equations.
– user121049
Sep 1 at 10:47





You've missed out $lambda_3$ from 4th and 5th equations.
– user121049
Sep 1 at 10:47
















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