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:
$\left( a \right)y' = {y^2}$
$\left( b \right)y'' = y'y$
$\left( c \right)yy'' = y'$
$\left( d \right)yy'' = {\left( {y'} \right)^2}$

Answer 1

Hint: In this particular type of question use the concept that if there are n number of constants in the family of the curve so we have to differentiate it n number of times so that all constants are eliminated so in the given family of the curve there are two constants so we have to differentiate it two times so use these concepts to reach the solution of the question.

Complete step-by-step solution:
Given family of curves
$y = {c_1}{e^{{c_2}x}}$................ (1)
Now as we see that there are two constants present in the family of curve so we have to differentiate the above family of curve twice so that the constants are eliminated completely so differentiate it w.r.t x we have,
$ \Rightarrow \dfrac{d}{{dx}}y = \dfrac{d}{{dx}}{c_1}{e^{{c_2}x}}$
$ \Rightarrow y' = {c_1}\dfrac{d}{{dx}}{e^{{c_2}x}}$, $\left[ {\because \dfrac{d}{{dx}}y = y'} \right]$
Now as we know that $\dfrac{d}{{dx}}{e^{ax}} = {e^{ax}}\dfrac{d}{{dx}}\left( {ax} \right) = a{e^{ax}}$ so use this property in the above equation we have,
$ \Rightarrow y' = {c_1}{c_2}{e^{{c_2}x}}$
Now from equation (1) substitute the value we have,
$ \Rightarrow y' = {c_2}y$............... (2)
Now again differentiate this equation we have,
$ \Rightarrow \dfrac{d}{{dx}}y' = \dfrac{d}{{dx}}{c_2}y$
$ \Rightarrow \dfrac{d}{{dx}}y' = {c_2}\dfrac{d}{{dx}}y$
$ \Rightarrow y'' = {c_2}y'$................... (3)
Now divide equation (3) by equation (2) so that constant ${c_2}$ is eliminated so we have,
\[ \Rightarrow \dfrac{{y''}}{{y'}} = \dfrac{{{c_2}y'}}{{{c_2}y}}\]
\[ \Rightarrow \dfrac{{y''}}{{y'}} = \dfrac{{y'}}{y}\]
\[ \Rightarrow yy'' = {\left( {y'} \right)^2}\]
So this is the required answer.
Hence option (d) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the basic differentiation property such as $\dfrac{d}{{dx}}{e^{ax}} = {e^{ax}}\dfrac{d}{{dx}}\left( {ax} \right) = a{e^{ax}}$ so using this property differentiate the given function as above and eliminate the constants as above, we will get the required answer.