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Joint Entrance Examination
Mathematics
Differential Equations
Question 342
Question 342 of 490
Medium
MCQ
The differential equation which represents the family of curves
y
=
c
1
e
c
2
x
,
y = {c_1}{e^{{c_2}x}},
y
=
c
1
e
c
2
x
,
where
c
1
{c_1}
c
1
, and
c
2
{c_2}
c
2
are arbitrary constants, is
A
\[y'' = y'y\]
B
\[yy'' = y'\]
C
\[yy'' = {\left( {y'} \right)^2}\]
D
\[y' = {y^2}\]
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Official solution
We have
y
=
c
1
e
c
2
x
y = {c_1}{e^{{c_2}x}}
y
=
c
1
e
c
2
x
\[ \Rightarrow y' = {c_1}{c_2}{e^{{c_2}x}} = {c_2}y\] \[ \Rightarrow {{y'} \over y} = {c_2}\] \[ \Rightarrow {{y''y\left( {y'} \right){}^2} \over {{y^2}}} = 0\] \[ \Rightarrow y''y = {\left( {y'} \right)^2}\]
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