$\begin{align} & y={{e}^{2x}}\cos 4x \\ & \Rightarrow y'=2{{e}^{2x}}\cos 4x-4{{e}^{2x}}\sin 4x=2{{e}^{2x}}\left( \cos 4x-2\sin 4x \right) \\ \end{align}$

$\begin{align} & y''=4{{e}^{2x}}\left( \cos 4x-2\sin 4x \right)+{{e}^{2x}}\left( -\sin 4x-8\cos 4x \right) \\ & =4{{e}^{2x}}\left( -3\cos 4x-4\sin 4x \right) \\ \end{align}$

Xét mệnh đề: $Ay+By'+Cy''=0,\forall \mathbb{R}$

$\Leftrightarrow {{e}^{2x}}\left[ \left( A+2B-12C \right)\cos 4x-\left( 4B+16C \right)\sin 4x \right]=0,\forall x\in \mathbb{R}$

$\Leftrightarrow \left\{ \begin{align} & A+2B-12C=0 \\ & 4B+16C=0 \\ \end{align} \right.$ $\Leftrightarrow \left\{ \begin{align} & A+2B-12C=0 \\ & B=-4C \\ \end{align} \right.\Leftrightarrow A-20C=0$

Chọn $A=20,C=1$ và$B=-4$

Ta có $20y-4y'+y''=0$