SUMMARY
The discussion focuses on the non-dimensionalisation of two differential equations, specifically \(\frac{dc}{dT} = \alpha - \mu c\) and \(\frac{dN}{dt} = \frac{rN}{h + rN} - bNC\). The user successfully non-dimensionalised the first equation using substitutions \(N = N_0 n\), \(C = C_0 c\), and \(t = t_0 T\). However, they encountered difficulties with the left-hand side of the second equation. A proposed solution involves dividing the numerator and denominator of \(\frac{rN}{h + rN}\) by \(h\) and substituting \(n = \frac{rN}{h}\), leading to the transformed equation \(\frac{h}{r}\frac{dn}{dt} = \frac{n}{1 + n} - \left(\frac{br}{h}\right)nC\).
PREREQUISITES
- Understanding of differential equations
- Familiarity with non-dimensionalisation techniques
- Knowledge of substitution methods in mathematical equations
- Basic grasp of mathematical notation and terminology
NEXT STEPS
- Research advanced non-dimensionalisation techniques in differential equations
- Learn about stability analysis in non-dimensionalised systems
- Explore numerical methods for solving differential equations
- Study the implications of parameter scaling in mathematical modeling
USEFUL FOR
Mathematicians, engineers, and researchers involved in mathematical modeling and analysis of differential equations, particularly those focusing on non-dimensionalisation techniques.