Question

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

Answer 1

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