Mathematics plays a fundamental role in the development of modern science and technology. Many real-world problems arising in physics, biology, engineering, environmental science, computer science, and medical research can be analysed and understood through mathematical modelling. Mathematical models simplify complex systems and express them in the form of equations, allowing scientists and researchers to study the behaviour of changing quantities with accuracy and clarity. Among the various mathematical tools, differential equations occupy significant place because thy describe relationships involving rates of change and evolving systems. Differential equations are widely used to model natural phenomena such as population growth, radioactive decay, heat transfer, and spread of diseases, motion of falling bodies, electrical circuits, and chemical reactions. In particular, first-order differential equations are highly useful in understanding growth and decay processes where the rate of change of a quantity is proportional to the quantity itslf. These equations help predict future behaviour based on present conditions and are therefore valuable in scientific planning and technological applications. This paper discusses the applications of mathematics in modern science and technology with special emphasis on differential equations and population growth models. The law of natural growth is introduced by assuming that the rate of increase of a population is directly proportional to the existing population. By forming and solving the corresponding first-order differential equation, and exponential growth model is obtained. The paper also presents a real-life application involving bacterial population growth to demonstrate how mathematical techniques are used for prediction and analysis. The study highlights how mathematics supports scientific discoveries, technological innovations, and decision-making processes in modern society. Form predicting epidemics such as Covid-19 to analysing ecological systems and resource planning, mathematical methods provide reliable and efficient solutions. Thus, mathematics serves as the backbone of modern science and technology and continues to contribute significantly to future advancements in research and innovation.