If y=a ( _ex)+b ( _ex) then x^2y^2+xy_1= - Vidyadip

MCQ

If $\text{y}=\text{a}\cos(\log_\text{e}\text{x})+\text{b}\sin(\log_\text{e}\text{x})$ then $\text{x}^2\text{y}^2+\text{xy}_1=$

A
$0$
B
$y$
✓
$-y$
D
None of these

✓

Answer

Correct option: C.

$-y$

$\text{y}=\text{a}\cos(\log_\text{e}\text{x})+\text{b}\sin(\log_\text{e}\text{x})$
$\Rightarrow\text{y}_1=-\text{a}\sin(\log_\text{e}\text{x})\frac{1}{\text{x}}+\text{b}\cos(\log_\text{e}\text{x})\frac{1}{\text{x}}$
$\Rightarrow\text{y}_2=\frac{-\text{a}\sin(\log_\text{e}\text{x})+\text{b}\cos(\log_\text{e}\text{x})}{\text{x}}$
$\Rightarrow\text{y}_2=\frac{-\text{a}(\log_\text{e}\text{x})-\text{b}\sin(\log_\text{e}\text{x})-\{-\text{a}\sin(\log_\text{e}\text{x})+\text{b}\cos(\log_\text{e}\text{x})\}}{\text{x}^2}$
$\Rightarrow\text{x}^2\text{y}_2=-\{\text{a}\cos(\log_\text{e}\text{x})+\text{b}\sin(\log_\text{e}\text{x})\}$
$-\{-\text{a}\sin(\log\text{x})+\text{b}\cos(\log_\text{e}\text{x})\}$
$\Rightarrow\text{x}^2\text{y}_2=-\text{y}-\text{xy}_1$
$\Rightarrow\text{x}^2\text{y}_2+\text{xy}_1=-\text{y}$

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