Question

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

• A. $y^{\prime \prime }=y^{\prime }y$
• B. $y{y}^{\prime \prime }=y^{\prime }$
• C. $y{y}^{\prime \prime }={\left(y^{\prime }\right)}^2$
• D. $y^{\prime }=y^2$

Answer 1

Answer: (C)

$y{y}^{\prime \prime }={\left(y^{\prime }\right)}^2$

$y=c_1{e}^{c2x}$
Differentiating w.r.t. x, we get
$y^{\prime }=c_1{c}_2{e}^{c2x}=c_{2}y$ (i)
Again differentiating w.r.t. x
$y^{\prime \prime }=c_2{y}^{\prime }$ (ii)
From (i) and (ii) upon division
$\frac{y^{\prime }}{y^{\prime \prime }}=\frac{y}{y^{\prime }}$ $\Rightarrow$ $y^{\prime \prime }y={\left(y^{\prime }\right)}^2$
Which is the desired differential equation of the family of curves.