<![CDATA[This report explores the fundamental use of differential equations in understanding and modeling dynamic systems, tracing its roots for the contributions of mathematicians. Differential equations act as a basic platform for scientific and engineering research, providing insights into the dynamics of physical, and social systems. Their adaptability and associative applicability, especially in fields like environmental science and technology learning, highlight their main importance. The report dwells with specific applications in engineering, emphasizing their role in dynamic systems, control theory, and optimization. The definitions and types of differential equations are explained, showcasing their diverse characteristics. The historical evolution of differential equations, spanning centuries, underscores their continual refinement and application in various scientific disciplines. Moreover, the report presents hypothetical case studies illustrating the application of differential equations in the calculation of mass of fuel tank of rocket, time required by rocket to become triple its initial velocity. These examples showcase the practical utility of differential equations in enhancing precision and efficiency in space exploration. The advantages of application of differential equations in three-dimensional space are highlighted, emphasizing their role in realistic modeling, multidimensional dynamics, and scientific exploration. However, the report also contains certain drawback, such as increased complexity, computational intensity, and visualization challenges associated with three-dimensional systems. In conclusion, the study of differential equations remains vital for unraveling the complexities of the natural world and technological advancements, demonstrating their enduring significance in advancing human knowledge, healthcare, and innovation.]]>
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