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- Thread starter lonewolf219
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In summary, the general solution of a 2nd order homogeneous equation is linear because it is a linear combination of two linearly independent solutions. The notation e^(xt) should actually be e^(rt) where r is found using the characteristic equation. The solution is considered linear because the function e^rt is linear in y, not x.

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lonewolf219 said:

The general solution is a linear combination of two linearly independent solutions y

p.s. I think something is not right with your notation e^(xt) .

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Homework Helper

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It is linear in y not x.

L is a linear operator if

L[Ʃa_{n}y_{n}]=Ʃa_{n}L[y_{n}]

L is a linear operator if

L[Ʃa

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Ah, OK. Thanks guys.

Understanding the linearity of general solutions of homogeneous equations is important because it allows us to solve for a wide range of variables and constants, making it a versatile tool in many areas of science and mathematics.

A homogeneous solution is one in which all terms have the same degree or power, meaning there are no constant terms. In other words, the solution is not affected by any changes in the independent variable.

A solution to a homogeneous equation is linear if it satisfies the properties of linearity, which include additivity and scalability. Additivity means that when two solutions are added together, the resulting solution is also a valid solution. Scalability means that when a solution is multiplied by a constant, the resulting solution is also a valid solution.

No, a non-linear equation cannot have a linear general solution. A linear general solution implies that the equation is linear, meaning that it follows the properties of linearity. If an equation is non-linear, it does not follow these properties and therefore cannot have a linear general solution.

We can use the linearity of general solutions to solve for unknown constants by plugging in different values for the variables and using the properties of linearity to manipulate the equation and solve for the unknown constants. This method is commonly used in solving systems of linear equations.

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