We substitute into the differential equation:
dG~idTa=j=14ij[ijG~i(Ta)G~j(Ta)+ijR(Ta)]\frac{d\widetilde{\mathcal{G}}_i}{dT_a} = \sum_{j=1}^4 \gamma_{ij} \left[ \alpha_{ij} \cdot \widetilde{\mathcal{G}}_i(T_a) \cdot \widetilde{\mathcal{G}}_j(T_a) + \beta_{ij} \cdot \mathcal{R}(T_a) \right]
3. Separate Components and Factorize
We factor the components:
dG~idTa=G~i(Ta)j=14ijijG~j(Ta)+j=14ijijR(Ta)\frac{d\widetilde{\mathcal{G}}_i}{dT_a} = \widetilde{\mathcal{G}}_i(T_a) \cdot \sum_{j=1}^4 \gamma_{ij} \alpha_{ij} \cdot \widetilde{\mathcal{G}}_j(T_a) + \sum_{j=1}^4 \gamma_{ij} \beta_{ij} \cdot \mathcal{R}(T_a)
Suppose we define two quantities:
Ai(Ta)=j=14ijijG~j(Ta)A_i(T_a) = \sum_{j=1}^4 \gamma_{ij} \alpha_{ij} \cdot \widetilde{\mathcal{G}}_j(T_a)
Bi(Ta)=j=14ijijR(Ta)B_i(T_a) = \sum_{j=1}^4 \gamma_{ij} \beta_{ij} \cdot \mathcal{R}(T_a)
So the equation becomes:
dG~idTa=G~i(Ta)Ai(Ta)+Bi(Ta)\frac{d\widetilde{\mathcal{G}}_i}{dT_a} = \widetilde{\mathcal{G}}_i(T_a)\cdot A_i(T_a) + B_i(T_a)
4. Solution Analysis: Modified Riccati Equation Type
