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 Question:
We must form a differential equation by eliminating two arbitrary constants ($c_1, c_2$) from the given family of curves. Since there are two constants, we will need to differentiate the equation exactly twice.

Step 2: Detailed Explanation:
The given equation is:
$y = c_1 e^{c_2 x}$ --- (Equation 1)
Differentiate both sides with respect to $x$:
$y' = c_1 c_2 e^{c_2 x}$ --- (Equation 2)
Notice that the term $c_1 e^{c_2 x}$ appears in the derivative, which is exactly equal to $y$. Substitute $y$ into Equation 2 to eliminate $c_1$:
$y' = c_2 y$
To make further differentiation easier and eliminate $c_2$, isolate it:
$\frac{y'}{y} = c_2$ --- (Equation 3)
Now, differentiate Equation 3 with respect to $x$ using the quotient rule:
$\frac{d}{dx} \left( \frac{y'}{y} \right) = \frac{d}{dx} (c_2)$
$\frac{y \cdot y'' - y' \cdot y'}{y^2} = 0$
Since $y \neq 0$ (exponential functions are non-zero), we can multiply both sides by $y^2$:
$y y'' - (y')^2 = 0$
Rearrange the terms:
$y y'' = (y')^2$

Step 3: Final Answer:
The differential equation is $yy'' = (y')^2$, matching option (c).