Question

The differential equation obtained by eliminating the arbitrary constant from the equation \[ y^2 = (2x+c)^5 \] is

• A. \(\left(\dfrac{dy}{dx}\right)^4 - 625y^4 = 0\)
• B. \(\left(\dfrac{dy}{dx}\right)^5 - 3125y^3 = 0\)
• C. \(\left(\dfrac{dy}{dx}\right)^3 - 125y^3 = 0\)
• D. \(xy\dfrac{dy}{dx} = 5\)

Hint:

To eliminate constants, differentiate and then substitute the original equation to remove the constant completely.

Answer 1

The Correct Option is B

Step 1: Differentiate the given equation.
Given: \[ y^2 = (2x+c)^5 \] Differentiate both sides w.r.t. \(x\): \[ 2y\frac{dy}{dx} = 5(2x+c)^4 \cdot 2 \] \[ 2y\frac{dy}{dx} = 10(2x+c)^4 \]
Step 2: Express \((2x+c)\) in terms of \(y\).
From the original equation: \[ (2x+c)^5 = y^2 \Rightarrow (2x+c) = y^{2/5} \]
Step 3: Substitute in the differentiated equation.
\[ 2y\frac{dy}{dx} = 10(y^{2/5})^4 \] \[ 2y\frac{dy}{dx} = 10y^{8/5} \]
Step 4: Simplify.
\[ \frac{dy}{dx} = 5y^{3/5} \]
Step 5: Eliminate fractional powers.
Raise both sides to the power 5: \[ \left(\frac{dy}{dx}\right)^5 = 3125y^3 \]
Step 6: Write the final differential equation.
\[ \left(\frac{dy}{dx}\right)^5 - 3125y^3 = 0 \]