# ODEs and linear algebra

Create a 3 page essay paper that discusses ODEs and Linear Algebra.by differential equations of first or higher order that must be generally or particularly solved to find a certain equation that would potentially describe the behaviour of an object or a particle based on its variable property or set of properties. This notion applies to an aspect in engineering that deals with the task of analysing vessel leaks as in formulating a mathematical model for a leaking process being undergone by a substance that flows out of a crack or fault in the tank so that the flow rate correspondingly affects the depth of such fluid in the tank. The scenario could be interpreted either for a varying volume or a varying liquid depth with time which is often assumed proportional to each other in a differential equation (MIT, n.d., 1).Use your own examples to show how to solve one first order, linear, ordinary differential equation where an integrating factor is needed in the solution. Explain why the integrating factor works and when should this method be used.*The method of using integrating factor renders the differential equation exact so that in this form, it would be more convenient to evaluate the differential equation and arrive at the solution.Use your own examples to explain to a first year Engineering student what are homogeneous linear equations and where they are found in engineering. Also explain the analytical methods that can solve them.By the term homogeneous, linear equations are illustrated in the simple sense of merely being composed of homogeneous differential terms, bearing derivatives of one variable type with respect to another on one side of the equation while the other side of the equation is kept as zero or one with constant term rather than another type of variable. This may be generally exemplified as (Dawkins, 2003  2014):Besides their applicability to mathematical analysis of a system that undergoes a leaking process, homogeneous linear equations are extensively utilized in modelling equations

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