Chapter 8 - Linear Differential Equations of Order n - 8.2 Constant Coefficient Homogeneous Linear Differential Equations - True-False Review - Page 513: a

False

Let us consider an example: $\dfrac{d^3y}{dx^3}-2\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}+8y=0$ Write auxiliary equation: $r^3-2r^2-4r+8=(r+2)(r-2)^2=0$ $r_1=-2; r_2=2$ So, we have three linearly independent solutions to the equation such as: $y_1=e^{-2x}, y_2(x)=e^{2x}$ and $y_3(x)=xe^{2x}$ Therefore, the given statement is False.