Question

The differential equation which represents the family of curves $y=c_1e^{c_2x}$, where $c_1$ and $c_2$ are arbitrary constants is

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

Answer 1

Answer: (D)

$yy ''=(y')^2$

Answer 2

$y=c_{1}e^{c_{2}x}\,...\left(1\right)$ $y '=c_{2}c_{1}e^{c_{2}x}$ $y '=c_{2}y \,...\left(2\right)$ $y ''=c_{2}y '$ From $\left(2\right)$ $c_{2}=\frac{y '}{y}$ So, $y "=\frac{\left(y '\right)^{2}}{y} \Rightarrow yy "=\left(y '\right)^{2}$