Consider the following statement:
“Ath-order constant-coefficient homogeneous linear equation haslinearly independent solutions if and only if the corresponding auxiliary polynomial hasdistinct roots.”
Verify this statement either true or false as follows:
Consider the 3rd order constant-coefficient homogeneous linear equation,
Its auxiliary polynomial is,
This implies, the roots of the polynomial are.
And, the distinct roots of the polynomial are.
So, the number of distinct roots of the polynomial is 2.
The general solution of the equationis,
Now, the Wronskian of the three solutionsis,
As the Wronskian is nonzero, the 3 solutionsare linearly independent. But the number of distinct roots of the polynomial is 2.
This result provides contradiction to the statement given above.
So, the provided statement is.