Question

The differential equation which represents the family of curves $y = c_1 e^{c_2 x}$, where $c_1, c_2$ are arbitrary constants is ______.

• A. $y'' = y' y$
• B. $yy' = y'$
• C. $yy'' = (y')^2$
• D. $y' = y^2$

Hint:

A faster method: Take the natural log first!
$\ln y = \ln(c_1) + c_2 x$
Differentiate once: $\frac{y'}{y} = c_2$.
Differentiate twice: The RHS becomes 0 immediately, yielding the quotient rule derivative equaling zero!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the differential equation, we must eliminate the arbitrary constants $c_1$ and $c_2$ by differentiating the equation as many times as there are constants (twice).

Step 2: Formula Application:
1. $y = c_1 e^{c_2 x}$ 2. $y' = c_1 c_2 e^{c_2 x} = c_2 y$

Step 3: Explanation:
From $y' = c_2 y$, we get $c_2 = \frac{y'}{y}$. Differentiating $y' = c_2 y$ again with respect to $x$: $y'' = c_2 y'$ Substitute the value of $c_2$: $y'' = \left( \frac{y'}{y} \right) y' \implies y'' = \frac{(y')^2}{y} \implies yy'' = (y')^2$.

Step 4: Final Answer:
The differential equation is $yy'' = (y')^2$.