Question

Which of the following functions are solutions of the differential equation y'' + y = sin(x)?a) y= sinx
b) y= cosx
c) y=1/2sinx
d) -1/2xcosx

Answer 1

Answer:

Option (d)

Step-by-step explanation:

Given,

y" +y=sin x ...........(1)

The particular solution

$y_p=A x sinx +Bx cosx$

$y'_p=Axcosx+Asinx+B cosx-Bxsinx$

$y"_p=Acosx-Axsinx+Acosx-Bsinx-Bsinx-Bxcosx$

$y"_p=2Acosx-Axsinx-2Bsinx-Bxcosx$

Putting the value of y" and y in equation (1)

$2Acosx-Axsinx-2Bsinx-Bxcosx+Axsinx+Bxcosx = sinx$

$\Rightarrow 2Acosx-2Bsinx=sinx$

Therefore 2A =0              -2B=1

              ⇒A=0                 $\rightarrow B=-\frac{1}{2}$

Therefore $y_p=-\frac{1}{2} x cosx$