This method determines the solution to the point reactor kinetics model for constant reactivity by using solutions to the linear matrix with constant entries. They can be extended to a recursive problem for non-constant reactivity. First you need to expand the neutron density and delayed neutron equations. These are then built into a set of linear first-order differential equations in matrix form.
The first term expansion of the point kinetics equations
\begin{matrix} \frac{d}{dt}n_j(t)=\frac{\rho_0-\beta}{\Lambda}n_j(t)+\sum^6_{i=1}\lambda_iC_{ij}(t)+\frac{\rho_1(t)}{\Lambda}n_{j-1}(t) \\ \frac{d}{dt}C_{1j}(t)=\frac{\beta_1}{\Lambda}n_j(t)-\lambda_1C_{1j}(t) \\ \vdots \\ \frac{d}{dt}C_{6j}(t)=\frac{\beta_6}{\Lambda}n_j(t)-\lambda_6C_{6j}(t) \\ \end{matrix}
Which gives the non-homogenous matrices
\begin{bmatrix} \frac{d}{dt}n_0 \\ \frac{d}{dt}C_{10} \\ \vdots \\ \frac{d}{dt}C_{60} \end{bmatrix}= \begin{bmatrix} \frac{\rho_0-\beta}{\Lambda} & \lambda_1 & \dots & \lambda_6 \\ \frac{\beta_1}{\Lambda} & -\lambda_1 & 0 & 0 \\ \vdots & 0 & \ddots & \vdots \\ \frac{\beta_6}{\Lambda} & 0 & \dots & -\lambda_7 \\ \end{bmatrix} \begin{bmatrix} n_0 \\ C_{10} \\ \vdots \\ C_{60} \end{bmatrix}
These matrices are then solved using the Laplace transform technique with this decomposition method for multiple eigenvalues. The nonhomogeneous equation is
\begin{bmatrix} \frac{d}{dt}n_0 \ \frac{d}{dt}C_{10} \ \vdots \ \frac{d}{dt}C_{60} \end{bmatrix}= \begin{bmatrix} \frac{\rho_0-\beta}{\Lambda} & \lambda_1 & \dots & \lambda_6 \\ \frac{\beta_1}{\Lambda} & -\lambda_1 & 0 & 0 \\ \vdots & 0 & \ddots & \vdots \\ \frac{\beta_6}{\Lambda} & 0 & \dots & -\lambda_7 \\ \end{bmatrix} \begin{bmatrix} n_0 \\ C_{10} \\ \vdots \\ C_{60} \end{bmatrix}+ \begin{bmatrix} \frac{\rho_1(t)}{\Lambda} & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\dots& 0 \end{bmatrix} \begin{bmatrix} n_{j-1} \\ C_{1j-1} \\ \vdots \\ C_{6j-1} \end{bmatrix}
[[Linear Differential Matrix Equations]]
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