Question

y''-7y'+12y=e^2x помогите решить

Answer 1

$y''-7y'+12y=e^{2x}\\\\1)\; \; k^2-7k+12=0\; ,\; \; k_1=4\; ,\; k_2=3\; \; (teorema\; Vieta)\\\\\underline {y_{obsh.odnor.}=y_{o}=C_1\cdot e^{4x}+C_2\cdot e^{3x}}\\\\2)\; \; y_{chastn.neodnor.}=\tilde{y}=A\cdot x^0\cdot e^{2x}=Ae^{2x}\; \; \; (\alpha =2\ne3\ne4\; \; \to \; \; r=0)\\\\\tilde{y}'=2Ae^{2x}\\\\\tilde{y}''=4Ae^{2x}\\\\\tilde{y}''-7\tilde {y}'+12\tilde {y}=4Ae^{2x}-14Ae^{2x}+12Ae^{2x}=e^{2x}\\\\2Ae^{2x}=e^{2x}\; \; \Rightarrow \; \; \; 2A=1\; \; ,\; \; A=\frac{1}{2}\\\\\underline {\tilde{y}=\frac{1}{2}e^{2x}}$

$3)\; \; \underline {\; y_{obsh.neodnor.}=y_{o}+\tilde{y}=C_1\cdot e^{4x}+C_2\cdor e^{3x}+\frac{1}{2}\, e^{2x}\; }$