Question

If the equation of family of curves be $y=a\cos (x+b)$,where a,b are arbitrary constants,and $y^{{}^{\prime }}=\frac{dy}{dx}$,$y^{{}^{\prime \prime }}=\frac{d^{2}y}{d{x}^{2^{{}^{\prime }}}}$ etc ,then their differential equation is :

• A. $y^{{}^{\prime \prime }}=y$
• B. $y^{{}^{\prime \prime }}=y{y}^{{}^{\prime }}$
• C. $y^{{}^{\prime \prime }}+y=0$
• D. $y^{{}^{\prime \prime }}=y+y^{{}^{\prime }}$

Answer 1

The correct option is C $y^{{}^{\prime \prime }}+y=0$

$y=a\cos (x+b)$ ...(1)
$\Rightarrow y^{{}^{\prime }}=-a\sin (x+b)$ ...(2)
$\Rightarrow y^{{}^{\prime \prime }}=-a\cos (x+b)$ ...(3)
Adding (1) and (3), we get
$y^{{}^{\prime \prime }}+y=0$