Question

The graph of the function $y=f\left(x\right)$ passes through the point $(0,1)$ and satisfies the differential equation $\frac{dy}{dx}+y\cos x=\cos x$. Then which of the following is/are true regarding $f\left(x\right)$:

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

Answer 1

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

It is continuous and differentiable for all $x$

$\frac{dy}{dx}+y\cos x=\cos x\left(\text{linear}\right)$
On comparing with $\frac{dy}{dx}+Py=Q,$
We get $P=\cos x$ and $Q=\cos x$
I.F. $=e^{\int \cos xdx}=e^{\sin x}$
$\therefore$ Solution is $y{e}^{\sin x}=\int e^{\sin x}\cos xdx=e^{\sin x}+c$

When $x=0,y=1$ then $c=0$
$\Rightarrow y=1.$
Clearly the function is a constant function and hence periodic. Also it is continuous and differentiable for all $x$.