\lambda = 3\;\;\;\;\;\;\;\lambda = 1=>y_{oo}=C_1e^{3x}+C_2e^x\\ y_{r_H}=Ax+B=>y'=A=>y''=0\\ -4A+3Ax+3B=x+2\\ A=\dfrac{1}{3}=>B=\dfrac{10}{9}\\ y_{r_H}=\dfrac{1}{3}x+\dfrac{10}{9}\\ y=y_{r_H}+y_{oo}=\dfrac{1}{3}x+\dfrac{10}{9}+C_1e^{3x}+C_2e^x" alt="y''-4y'+3y=x+2\\ \lambda^2-4\lambda+3=0=>\lambda = 3\;\;\;\;\;\;\;\lambda = 1=>y_{oo}=C_1e^{3x}+C_2e^x\\ y_{r_H}=Ax+B=>y'=A=>y''=0\\ -4A+3Ax+3B=x+2\\ A=\dfrac{1}{3}=>B=\dfrac{10}{9}\\ y_{r_H}=\dfrac{1}{3}x+\dfrac{10}{9}\\ y=y_{r_H}+y_{oo}=\dfrac{1}{3}x+\dfrac{10}{9}+C_1e^{3x}+C_2e^x" align="absmiddle" class="latex-formula">