Question

Решить дифференциальное уравнение y"-12y'+36y=14e^6x

Answer 1

$y''-12y'+36y=14e^{6x},\ lambda^2-12lambda+36=0,\ lambda_{1,2}=6.\ y_o(x)=e^{6x}(C_1+C_2x).\ tilde{y}(x)=Ax^2e^{6x},,tilde{y}'(x)=Ae^{6x}(6x^2+2x),\ tilde{y}''(x)=Ae^{6x}(36x^2+24x+2),\ tilde{y}''-12tilde{y}'+36tilde{y}=14e^{6x}Leftrightarrow\ Aleft(36x^2+24x+2-12(6x^2+2x)+36x^2right)=14,,A=7.\ tilde{y}=7x^2e^{6x} y(x)=y_o+tilde{y}=e^{6x}(C_1+C_2x+7x^2).$