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Question

If the solution of y=xdydx+dydx(dydx)2;d2ydx20 is y=f(x) and it is defined as f:RR then f(x) is

A
one-one
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B
into
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C
even
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D
onto
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Solution

The correct option is B into
y=xdydx+dydx(dydx)2
Differentiating w.r.t. x,
y=xy′′+y+y′′2yy′′0=(x+12y)y′′y=x+12
Putting it in the given differential equation we get,
y=(x+1)×(x+1)2(x+1)24y=(x+1)24
So,
f(x)=(x+1)24
The function is many one.

As f(x)f(x) so, it is not even.

The range of function is [0,)
Therefore, the function is into.

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