Question

The differential equation of minimum order by eliminating the arbitrary constants $A$ and $C$ in the equation $y=A\left[\sin \right(x+C)+\cos (x+C\left)\right]$ is

• A. $y"+(\sin x+\cos x)y^{\prime }=1$
• B. $y"+(\sin x+\cos x)y^{\prime }$
• C. $y"=(y^{\prime }{)}^2+\sin x\cos x$
• D. $y"+y=0$

Answer 1

Answer: (D)

$y"+y=0$

$y=A\left[\sin \right(x+C)+\cos (x+C\left)\right]=A[\sin x\cos c+\sin c\cos x+\cos x\cos c-\sin x\sin c]=\sin x(A\cos c-A\sin c)+\cos x(A\sin c-A\cos c)y^{\prime }=\cos x(A\cos c-A\sin c)-\sin x(A\sin c-A\cos c)y^{\prime \prime }=-\sin x(A\cos c-A\sin c)-\cos x(A\sin c-A\cos c)=-y{y}^{\prime \prime }+y=0$

So correct answer will be option D