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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′′=yy
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B
yy′′=(y)2
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C
yy′′=y
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D
y=y2
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Solution

The correct option is B $$yy ^ {'' } = \left( y ^ {'} \right) ^ { 2 }$$
$$y={ c }_{ 1 }{ e }^{ { c }_{ 2 }x }$$
Now,
$$\Rightarrow \dfrac { dy }{ dx } ={ c }_{ 1 }{ c }_{ 2 }{ e }^{ { c }_{ 2 }x }$$
$$\Rightarrow { y }^{ ' }={ c }_{ 2 }y\quad \longrightarrow \left( 1 \right) $$
$${ y }^{ " }={ c }_{ 2 }{ y }^{ 1 }\quad \longrightarrow \left( 2 \right) $$
From $$(1)$$ we have, $${ c }_{ 2 }=\dfrac { { y }^{ ' } }{ y } $$, Substituting in $$(2)$$ we get
$$\Rightarrow { y }^{ " }=\dfrac { { \left( { y }^{ ' } \right)  }^{ 2 } }{ y } $$
$$\Rightarrow { yy }^{ " }={ \left( { y }^{ 1 } \right)  }^{ 2 }$$
Answer : Option B.

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