Are the particular integral and complementary function solutions of a Differential Equation linearly independent?

1 Answer
Steve M
Dec 16, 2016

Yes, Because of the Principle of Superposition

Explanation:

Yes the solutions #y_c# and #y_p# must be linearly independent

Why? Because of the Principle of Superposition

If it is known that the solutions #y_1#, #y_2#.....#y_n#, in #y_c#, are fundamental set of solutions to the homogeneous equation, and are linearly independent then from the Principle of Superposition

#y_("sup") = c_1y_1 + c_2y_2 + ... c_ny_n# where #c_1, c_1, ... c_n# are constants is also a solution of the homogeneous equation.

So then it follows that if #y_c# and #y_p# were not linearly independent then #y_p# could be written as superposition of the existing solutions that form #y_c#.

Therefore it would be a solution of the homogeneous equation, and therefore it would not be the general solution of the non-homogeneous equation

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