Question

The differential equation corresponding to the family of curves $y=A\cos (Bx+D),$ where $A,$ $B$ and $D$ are parameters, is

• A. of order $3$
• B. of order $2$
• C. of degree $2$
• D. of degree $1$

Answer 1

Answer: (A), (D)

of degree $1$

$y=A\cos (Bx+D)$
$\Rightarrow y=A[\cos Bx\cos D-\sin Bx\sin D]$
$\Rightarrow y=C_1\cos Bx+C_2\sin Bx,$ where $C_1=A\cos D,C_2=-A\sin D$
Differentiating w.r.t. $x,$
$y_1=-B{C}_1\sin Bx+B{C}_2\cos Bx$
Again differentiating again w.r.t. $x,$
$y_2=-B^2{C}_1\cos Bx-B^2{C}_2\sin Bx$
$\Rightarrow y_2=-B^2(C_1\cos Bx+C_2\sin Bx)$
$\Rightarrow y_2=-B^{2}y$
$\Rightarrow \frac{y_2}{y}=-B^2$
Again differentiating again w.r.t. $x,$
$\frac{y\cdot y_3-y_2\cdot y_1}{y^2}=0$
$\Rightarrow y\frac{d^{3}y}{d{x}^3}=\frac{dy}{dx}\cdot \frac{d^{2}y}{d{x}^2}$
Hence, order $=3$ and degree $=1$

Order can also be determined by the number of independent parameters.
Here $A,$ $B$ and $D,$ all three are independent, hence order is $3.$