Chapter 8 - Linear Differential Equations of Order n - 8.8 A Differential Equation with Nonconstant Coefficients - True-False Review - Page 567: c

False

For $y=x^r$ $r(r-1)+9r+16=0 \implies (r+4)(r+4)=0$ So, $r_1=-4;r_2=-4$ Thus, $x^{-4}$ and $x^{-4}$ are linearly independent solution which does not have the form of $x^r$. Therefore, the given statement is $\bf{False}$.