The sum YI + Y2 of two solutions YI and Y2 of the homogeneous equation (2) is a solution
Chapter 1, Problem 1.1.139(choose chapter or problem)
These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear ones, which have no similar properties.
Show that nonhomogeneous linear ODEs (1) and homogeneous linear ODEs (2) have the following properties. Illustrate each property by a calculation for two or three equations of your choice. Give proofs.
The sum \(y_{1}+y_{2}\) of two solutions \(y_{1}\) and \(y_{2}\) of the homogeneous equation (2) is a solution of (2), and so is a scalar mUltiple \(ay_{1}\) for any constant a. These properties are not true for (1)!
Text Transcription:
y_1 + y_2
y_1
y_2
ay_1
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