Question

Решить дифференциальное уравнение: y'' - y = x* cosx

Answer 1

$y'' - y = x* cosx\\\lambda^2-1=0\\\lambda^2=1\\\lambda=^+_-1\\Y=C_1e^{x}+C_2e^{-x}\\\hat{y}=Axcosx+Bcosx+Cxsinx+Dsinx\\\hat{y}'=Acosx-Axsinx-Bsinx+Csinx+Cxcosx+Dcosx\\\hat{y}''=-2Asinx-Axcosx-Bcosx+2Ccosx-Cxsinx-Dsinx\\\\-2Asinx-Axcosx-Bcosx+2Ccosx-Cxsinx-Dsinx-\\-Axcosx-Bcosx-Cxsinx-Dsinx=xcosx\\-2Asinx-2Axcosx-2Bcosx+2Ccosx-2Cxsinx-\\-2Dsinx=xcosx\\sinx|-2A-2D=0=\ \textgreater \ D=\frac{1}{2}\\cosx|-2B+2C=0=\ \textgreater \ B=0\\xsinx|-2C=0=\ \textgreater \ C=0\\xcosx|-2A=1=\ \textgreater \ A=-\frac{1}{2}\\\hat{y}=-\frac{x}{2}cosx+\frac{1}{2}sinx$
$y=Y+\hat{y}=C_1e^{x}+C_2e^{-x}-\frac{x}{2}cosx+\frac{1}{2}sinx$