Answer to Question #225917 in Chemical Engineering for Lokika

Question

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

Answer 1

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

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

$D^2+4=0⟹D=±2i$

$∴C.F.=C_1cos2x+C_2sin2x$

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

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