Question

The graph of the function $y=f\left(x\right)$ passing through the point (0, 1) and satisfying the differential equation $\frac{dy}{dx}+ycosx=cosx$ is such that:

• A. it is a constant function
• B. it is periodic
• C. it is neither an even nor an odd function
• D. it is continuous & differentiable for all $x$

Answer 1

Answer: (A), (B), (D)

The correct options are
A it is periodic
B it is a constant function
D it is continuous & differentiable for all $x$

$\frac{dy}{dx}+y\cos x=\cos x\Rightarrow \frac{dy}{dx}=\cos x(1-y)\Rightarrow \frac{\frac{dy}{dx}}{(1-y)}=\cos x\Rightarrow \int \frac{\frac{dy}{dx}}{(1-y)}dx=\int \cos xdx\Rightarrow -\log (1-y)=\sin x+c\Rightarrow y=-\frac{c}{e^{\sin x}}+1$
As it passes through $(0,1)$
$1=-c+1\Rightarrow c=0$
Gives $y=1$