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Formation of a Differential Equation from a General Solution

The function ...

Question

The function defined by the equation $x y - log y = 1$ satisfies $x (y y^{''} + y^{' 2}) - y^{''} + k y y^{'} = 0$ . Find the value of $k$ .

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Solution

The correct option is C $3$
Given, equation is
$x y - log y = 1$
On differentiating it w.r.t.x, we get
$x y^{'} + y \cdot 1 - \frac{1}{y} \cdot y^{'} = 0$
$\Rightarrow x y y^{'} + y^{2} - y^{'} = 0$
Again, differentiating w.r.t. $x$ , we get
$((x y - 1) y^{''} + y^{'} (x y^{'} + y \cdot 1) + 2 y y^{'} = 0$
$\Rightarrow x (y y " + y^{' 2}) - y^{''} + 3 y y^{'} = 0$
Comparing it with the given equation, we get $k = 3$

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