Question

The differential equation of the family of curves \[ y=ce^{2x} \] is:

• A. \(\frac{dy}{dx}=2y\)
• B. \(\frac{dy}{dx}=y\)
• C. \(\frac{d^2y}{dx^2}=2y\)
• D. \(\frac{dy}{dx}=2x\)

Hint:

To form a differential equation, differentiate the given family and eliminate all arbitrary constants.

Answer 1

The Correct Option is A

Concept: The differential equation corresponding to a family of curves is obtained by eliminating the arbitrary constant from the given equation. Given family: \[ y=ce^{2x} \] where \(c\) is an arbitrary constant.

Step 1: Differentiate the given equation Differentiating both sides with respect to \(x\), \[ \frac{dy}{dx} = c\frac{d}{dx}(e^{2x}) \] Using chain rule, \[ \frac{dy}{dx} = c(2e^{2x}) \] \[ \frac{dy}{dx} = 2ce^{2x} \]

Step 2: Eliminate the arbitrary constant From the original equation, \[ y=ce^{2x} \] Substituting into the differentiated equation, \[ \frac{dy}{dx} = 2y \]

Step 3: Write the required differential equation Hence the differential equation representing the family is \[ \boxed{\frac{dy}{dx}=2y} \] Final Answer: \[ \boxed{\frac{dy}{dx}=2y} \]