Question #223867

Solve d^{2}y/dx^{2}+2dy/dx+y=xsinx

Expert's answer

The characteristic equation of the homogeneous DE

"k^2+2k+1=0"Roots

"k_1=k_2=-1"Hence, the general solution of homogeneous DE

"y_h(x)=C_1e^{-x}+C_2xe^{-x}"The partial solution of inhomogeneous DE

"y_i(x)=(A+Bx)\\sin x+(C+Dx)\\cos x"We get

"y'_i(x)=(A+Bx)\\cos x-(C+Dx)\\sin x\\\\+B\\sin x+D\\cos x""y''_i(x)=-(A+Bx)\\sin x-(C+Dx)\\cos x\\\\+2B\\cos x-2D\\sin x"

"-(A+Bx)\\sin x-(C+Dx)\\cos x\\\\+2B\\cos x-2D\\sin x+2((A+Bx)\\cos x\\\\-(C+Dx)\\sin x+B\\sin x+D\\cos x)\\\\+(A+Bx)\\sin x+(C+Dx)\\cos x\\\\=x \\sin x"

So, the partial solution of inhomogeneous DE

"y_i(x)=\\frac{1}{2}(\\cos x-x\\cos x+\\sin x)"Finally,

"y(x)=C_1e^{-x}+C_2xe^{-x}\\\\+\\frac{1}{2}(\\cos x-x\\cos x+\\sin x)"

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