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

Answer 1

The correct option is B $y{y}^{{}^{\prime \prime }}={\left(y^{{}^{\prime }}\right)}^2$

$y=c_1{e}^{c_{2}x}$

Now,

$\Rightarrow \frac{dy}{dx}=c_1{c}_2{e}^{c_{2}x}$

$\Rightarrow y^{{}^{\prime }}=c_{2}y⟶\left(1\right)$

$y^{\prime \prime }=c_2{y}^1⟶\left(2\right)$

From $\left(1\right)$ we have, $c_2=\frac{y^{{}^{\prime }}}{y}$, Substituting in $\left(2\right)$ we get

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

$\Rightarrow {yy}^{\prime \prime }={\left(y^1\right)}^2$

Answer : Option B.