Question

The degree of the differential equation whose solution is $y^2 = 8a(x+a)$, is

• A. 2
• B. 1
• C. 4
• D. 3

Hint:

When a parameter $a$ appears linearly as well as quadratically ($a^2$), substituting its first-derivative value back into the equation will inherently create a term containing $\left(\frac{dy}{dx}\right)^2$. Thus, you can instantly predict that the degree will be 2 without finishing the algebraic simplification!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a general solution of a family of curves, $y^2 = 8a(x+a)$, where $a$ is an arbitrary constant parameter. We need to construct the corresponding differential equation by eliminating this parameter and then find its degree.

Step 2: Key Formula or Approach:
The order of a differential equation equals the number of independent arbitrary constants present in its general equation, which is 1 here (only $a$). The degree of a differential equation is the highest power (exponent) of the highest-order derivative present in it, after the equation has been cleared of fractional powers and radicals regarding derivatives.

Step 3: Detailed Explanation:
Let's expand the given equation: $$y^2 = 8ax + 8a^2 \quad \text{--- (Equation 1)}$$ Differentiate both sides with respect to $x$: $$2y\frac{dy}{dx} = 8a \implies a = \frac{2y}{8}\frac{dy}{dx} = \frac{y}{4}\frac{dy}{dx}$$ Now, substitute this expression for $a$ back into Equation 1 to eliminate the parameter: $$y^2 = 8\left(\frac{y}{4}\frac{dy}{dx}\right)x + 8\left(\frac{y}{4}\frac{dy}{dx}\right)^2$$ $$y^2 = 2xy\frac{dy}{dx} + 8\left(\frac{y^2}{16}\right)\left(\frac{dy}{dx}\right)^2$$ $$y^2 = 2xy\frac{dy}{dx} + \frac{y^2}{2}\left(\frac{dy}{dx}\right)^2$$ Multiply the entire equation by 2 to clear fractions: $$2y^2 = 4xy\frac{dy}{dx} + y^2\left(\frac{dy}{dx}\right)^2$$ In this differential equation, the highest-order derivative is $\frac{dy}{dx}$ (Order = 1). The highest power raised on this term is 2. Therefore, the degree of the differential equation is 2.

Step 4: Final Answer:
The degree of the differential equation is 2, which corresponds to option (A).