Question

The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$, where $c$ is a positive perimeter, is of

• A. Order $1$, degree $3$
• B. Order $2$, degree $3$
• C. Degree $1$, order $2$
• D. $Noneofthese$

Answer 1

The correct option is A Order $1$, degree $3$

Given family of curves
$y^2=2c(x+\sqrt{c})$ ...(i)

On differentiating both sides, we get
$2y\frac{dy}{dx}=2c(1+0)\Rightarrow c=y\frac{dy}{dx}$

From equation (i), we have

$y^2=2y\frac{dy}{dx}\{x+{\left(y\frac{dy}{dx}\right)}^{1/2}\}$

$\Rightarrow (y^2-2xy\frac{dy}{dx})=2{\left(y\frac{dy}{dx}\right)}^{3/2}$

On squaring both sides, we get
${(y^2-2xy\frac{dy}{dx})}^2=4{\left(y\frac{dy}{dx}\right)}^3$

So,

Order=$1$ (Order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation.)

Degree=$3$ (The degree of differential equation is represented by the power of the highest order derivative in the given differential equation.)