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Q: Is there any system which involves one nonlinear equation and one linear equation?

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linear system is like a chemistry equation or math equation where on both sides it must balance. Nonlinear is a math equation or physics that does not appear to have a direct answer just like chaos theory. lulu254ever

Linear system follows principal of superposition and homogeneity and Non linear system does not follow the same.

Wilson J. Rugh has written: 'Nonlinear system theory' -- subject(s): System analysis, Nonlinear theories 'Linear system theory' -- subject(s): Control theory, Linear systems

It is a system of linear equations which does not have a solution.

We study the nonlinear systems to understand the behavior of some circuit elements in the transient state.

A linear equation in n-dimensional space is of the form a1x1 + a2x2 + ... + anxn + c = 0 where the ai are numerical constants and the xi are variables. The equation represents a straight line in n-dimensional space. A non-linear equation is one in which one or more of the xi have a power other than 1. The equation will represent a curve. A linear system is one in which you will get the same result if you change an input by the same amount - from whatever starting level. Otherwise the system is non-linear.

No....not necessary

In general, a system of non-linear equations cannot be solved by substitutions.

When its matrix is non-singular.

a linear equation

In general all systems are nonlinear but we simplify this nonlinear vibration to linear ones so that we can get approximate results. Approximate results are still good results in many cases. For example when you analyze the vibrations of the simple pendulum for small vibrations you don't need to include aerodynamic drag which is a nonlinear in its nature. By neglecting the nonlinear parts we can derive the second order differential equations which describes the motion of the system in this case gives linear vibration of simple pendulum. Another good example would be an examination of system which consists of block of mass m, spring with stiffness k and viscous damper with damping coefficient c and let's say that the block of mass m is in contact with the surface. Now the spring stiffness and the viscous damping are in reality nonlinear but for small vibration we assume they are linear. The bloc of mass m is in contact with the surface so that means that between the block and the surface is a friction. So if we analyze this system with nonlinear terms we would need to include the nonlinear stiffness, nonlinear damping coefficient and nonlinear friction. These would result in the time consuming calculation and in the end the results would little more precise than the approximation. In nonlinear analysis we attack the differential equation which describes the motion of nonlinear system with small parameter and with this we expand the solution. This method is called perturbation method. To solve nonlinear systems you need to use specific perturbation method and these methods are: Straightforward expansion, domain perturbation, multiple scale analysis etc. For more information check my site Linear Vibration.

Linear systems are easier to understand and help you build an understanding of the workings of a system. Once you have a firm understanding of linear systems and the mathematics are understood you will be in a better position to understand more complex non-linear systems.

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