If y=y(x) is solution of differential equation (d y/d x)=-x sin x ; y(0)=0, then

Question

If y=y(x) is solution of differential equation (d y/d x)=-x sin x ; y(0)=0, then (A) y>0 if x ∈((-π/2), 0). (B) y<0 if x ∈((-π/2), 0). (C) number of inflection points of y=y(x) in (0, (π/2)) is 0 . (D) (d2 y/d x2)+y=-2 sin x

Tardigrade Admin · Accepted Answer

d y=-x sin x ⋅ d x y=x cos x- sin x+c y=x cos x- sin x Θ c=0 = cos x(x- tan x) y prime prime=x cos x- sin x=-(x cos x+ sin x)<0 text in (0, (π/2)) ⇒ text No inflection point.

Q. If $y = y (x)$ is solution of differential equation $\frac{d y}{d x} = - x sin x; y (0) = 0$ , then

$y > 0$ if $x \in (\frac{- π}{2}, 0)$ .

$y < 0$ if $x \in (\frac{- π}{2}, 0)$ .

number of inflection points of $y = y (x)$ in $(0, \frac{π}{2})$ is 0 .

$\frac{d ^{2} y}{d x ^{2}} + y = - 2 sin x$

$d y = - x sin x \cdot d x$
$y = x cos x - sin x + c$
$y = x cos x - sin x {Θ c = 0}$
$= cos x (x - tan x)$
$y^{''} = x cos x - sin x = - (x cos x + sin x) < 0 in (0, \frac{π}{2})$
$\Rightarrow No inflection point.$

Q. If y=y(x) is solution of differential equation dxdy​=−xsinx;y(0)=0, then

y>0 if x∈(2−π​,0).

y<0 if x∈(2−π​,0).

number of inflection points of y=y(x) in (0,2π​) is 0 .

dx2d2y​+y=−2sinx