Question #225917

The particular integral value of (D² + 4)y = sin²x is

A) 1/8-x/8 sin2x

B) 1/8+x/8 Sin2x

C) 1/8-x/8 Cos2x D)1/8+x/8 Cos2x

Expert's answer

"(D^2+4)y=sin^2x"

complementary function (C.F.) is obtained by putting

"D^2+4=0\u27f9D=\u00b12i"

"\u2234C.F.=C_1cos2x+C_2sin2x"

Now, a particular integral (P.I.) is found out as follows

"P.I.=\\frac{sin^2x}{D^2+4}\\\\\n=\\frac{\\frac{1\u2212cos2x}{2}}{D2+4}\\\\\n=\\frac{1}{2}(\\frac{1\u2212cos2x}{D^2+4})\\\\\n=\\frac{1}{2}(\\frac{1}{D^2+4}\u2212\\frac{cos2x}{D^2+4})\\\\\n=\\frac{1}{2}(\\frac{1}{0^2+4}\u2212\\frac{x}{2} \\frac{1}{D}(cos2x))\\\\\n=\\frac{1}{2}(\\frac{1}{4}\u2212\\frac{x}{2}\\frac{sin2x}{2})\\\\\n=\\frac18\u2212\\frac{xsin2x}{8}"

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