Why multiply possible form of part of particular solution form by a power of the independent variable when solving an ordinary differential equation

When solving a fixed-constant linear ordinary differential equation where the part of the homogeneous solution is same form as part of a possible particular solution, why do we get the next independent solution in the form of x^n* possible form of part of particular solution?  Show this through an example.

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Using Taylor polynomial to approximately solve an ordinary differential equation

Taylor polynomial is an essential concept in understanding numerical methods. Examples abound and include finding accuracy of divided difference approximation of derivatives and forming the basis for Romberg method of numerical integration.

In this example, we are given an ordinary differential equation and we use the Taylor polynomial to approximately solve the ODE for the value of the dependent variable at a particular value of the independent variable. As a homework assignment, do the following.
1) compare the approximate solution with the exact one, and
2) get another approximate solution by using a third order Taylor polynomial.

Taylor polynomial approximation of solving ordinary differential equations

You can visit the above example by opening a pdf or video file.

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