Answer to Question #224865 in Chemical Engineering for Lokika

Question #224865

Solve d²y/dx²+2 dy/dx+y=x sinx.


1
Expert's answer
2021-08-16T02:20:38-0400

"\\dfrac{d\u00b2y}{dx\u00b2}+2 \\dfrac{dy}{dx}+y=x \\sin x"


"(D\u00b2+2D+1)y=x\\sin x"


"A.E \\ is\\ m\u00b2+2m+1 =0\\\\\n(m+1)\u00b2= 0\\\\\nm=-1;\\ m=-1"


"P.I. =\\dfrac1{D\u00b2+2D+1}x \\sin x"


"=\\dfrac1{(D+1)\u00b2}x \\sin x;\\quad D\\implies D-1"


"=\\dfrac1{D\u00b2} x\\sin x= \\dfrac1D \\int{x \\sin x}\\ dx"


"=\\dfrac1D(-x\\cos x-\\sin x)"


Hence P.I. = "-2\\cos x-2x\\sin x"


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