Application of Variable Separation Technique in the Analysis of First-Order Differential Equations

Abstract

First order differential equations are used to model things that change at a rate that depends on the thing itself. We looked at three models: how a population grows, how radioactive things decay and Newton’s Law of Cooling. What we found out is that we can use a technique called separation of variables, on each of these models. This technique lets us transform the models into a form so we can integrate the variables on their own. The population growth solutions we got show that population growth happens fast when the growth rate is always the same. This is like what happens with things they get weaker really fast over time. When things cool down the difference in temperature gets smaller and smaller compared to the temperature around them. These results tell us that the method we split the problems not offers us clear solutions but also helps us comprehend the math behind population expansion and these other physical things, including radioactive chemicals and cooling processes. However, the study also identifies a limitation: the method is effective only when the differential equation can be represented in separable form. Equations involving non-separable terms require other approaches. Despite this restriction, the findings imply that separation of variables gives a systematic and transparent framework for solving a major class of first-order equations and acts as a conceptual bridge into more advanced analytical methods.