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

True

Let us counter an example: $(D^2-4D+5)^2y=0$ The four linearly independent solutions to equation are: $y_1(x)=e^{2x}cos x $, $y_{2x}=e^{2x}\sin x $, $y_3(x)=e^{2x}x\cos x$ and $y_4(x)=e^{2x} x \sin x$ The general solution is: $y(x)=c_1e^{2x}cos x +c_2e^{2x}sin x+c_3 e^{2x}x\cos x+c_4 e^{2x}\sin x$ Therefore, the given statement is true.