Question

The differential equation which represents the family of curves $y=c_1{e}^{c_{2}x}$ where $c_1$ and $c_2$ are arbitrary constants, is:

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

Answer 1

The correct option is D $y{y}^{\prime \prime }=(y^{\prime }{)}^2$

$y=c_1{e}^{c_{2}x}$ ... (i)
$\Rightarrow y^{\prime }=c_1{c}_2{e}^{c_{2}x}$
$\Rightarrow y^{\prime }=c_{2}y\dots$.. (from $\left(i\right)]$ ... (ii)
$\Rightarrow y^{\prime \prime }=c_2{y}^{\prime }$ (iii)
From (ii) \&(iii) $\frac{y^{\prime }}{y^{\prime \prime }}=\frac{y}{y^{\prime }}=(y^{\prime }{)}^2$
Hence, option 'D' is correct.