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Q.

The differential equation which represent the family of curves $y = a e^{b x}$ , where a and b are arbitrary constants, is

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a

$y^{'} = y^{2}$

b

$y y^{''} = y^{'}$

c

$y y^{''} = {(y^{'})}^{2}$

d

$y^{''} = y y^{'}$

answer is D.

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Detailed Solution

Given : $y = a e^{b x}$

$\ln y = \ln a + b x$

Differentiating twice, we get

$\frac{1}{y} y^{'} = b, \frac{y^{''}}{y} - \frac{1}{y^{2}} {(y^{'})}^{2} = 0 \Rightarrow y y^{''} = {(y^{'})}^{2} .$

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The differential equation which represent the family of curves y = aebx, where a and b are arbitrary constants, is