Question

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

• A. order 1, degree 4
• B. order 2, degree 3
• C. order 2, degree 4
• D. order 1, degree 3

Hint:

Remember to eliminate the parameter '$c$' by differentiating the given equation. The order of a differential equation is the order of the highest derivative present, and the degree is the highest power of the highest order derivative after making the equation free from radicals and fractions of derivatives.

Answer 1

The Correct Option is D

Step 1: Write down the given equation. The given family of curves is: \[ y^2 = 2c(x + \sqrt{c}) \quad \text{.... (i)} \]
Step 2: Differentiate the equation with respect to $x$ to eliminate the parameter $c$. Differentiating both sides of equation (i) with respect to $x$, we get: \[ \frac{d}{dx}(y^2) = \frac{d}{dx}(2c(x + \sqrt{c})) \] Since $c$ is a parameter, $\sqrt{c}$ is also a constant. So, $\frac{d}{dx}(\sqrt{c}) = 0$. \[ 2y \frac{dy}{dx} = 2c(1 + 0) \] \[ 2y \frac{dy}{dx} = 2c \] \[ c = y \frac{dy}{dx} \quad \text{.... (ii)} \]
Step 3: Substitute the expression for $c$ from (ii) into (i). Substitute $c = y \frac{dy}{dx}$ into equation (i): \[ y^2 = 2\left(y \frac{dy}{dx}\right)\left(x + \sqrt{y \frac{dy}{dx\right) \]
Step 4: Simplify the resulting differential equation. Divide both sides by $y$ (assuming $y \neq 0$): \[ y = 2\frac{dy}{dx}\left(x + \sqrt{y \frac{dy}{dx\right) \] \[ y = 2x \frac{dy}{dx} + 2\frac{dy}{dx} \sqrt{y \frac{dy}{dx \] Rearrange the terms to isolate the radical: \[ y - 2x \frac{dy}{dx} = 2\frac{dy}{dx} \sqrt{y \frac{dy}{dx \] Square both sides to eliminate the square root: \[ \left(y - 2x \frac{dy}{dx}\right)^2 = \left(2\frac{dy}{dx} \sqrt{y \frac{dy}{dx\right)^2 \] \[ \left(y - 2x \frac{dy}{dx}\right)^2 = 4\left(\frac{dy}{dx}\right)^2 \left(y \frac{dy}{dx}\right) \] \[ \left(y - 2x \frac{dy}{dx}\right)^2 = 4y \left(\frac{dy}{dx}\right)^3 \]
Step 5: Determine the order and degree of the differential equation. The highest order derivative present in the equation is $\frac{dy}{dx}$, which is a first-order derivative. Therefore, the order of the differential equation is 1. The highest power of the highest order derivative $\left(\frac{dy}{dx}\right)$ after making the equation free from radicals is 3. Therefore, the degree of the differential equation is 3. The differential equation is of order 1 and degree 3.