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

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

${y}_{\text{sup}} = {c}_{1} {y}_{1} + {c}_{2} {y}_{2} + \ldots {c}_{n} {y}_{n}$ where ${c}_{1} , {c}_{1} , \ldots {c}_{n}$ 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