If d y d x =y 2 x, y(0)=1, then solution is - Vidyadip

MCQ

If $\frac{d y}{d x}=y \sin 2 x, y(0)=1$, then solution is

✓
$y=e^{\sin ^2 x}$
B
$y=\sin ^2 x$
C
$y=\cos ^2 x$
D
$y=e^{\cos ^2 x}$

✓

Answer

Correct option: A.

$y=e^{\sin ^2 x}$

(a) : We have, $\frac{d y}{d x}=y \sin 2 x$
$
\Rightarrow \quad \frac{d y}{y}=\sin 2 x d x \Rightarrow \log y=-\frac{\cos 2 x}{2}+C
$
Since $y(0)=1 \Rightarrow x=0, y=1 \Rightarrow C=1 / 2$
$
\therefore \quad \log y=\frac{1}{2}(1-\cos 2 x) \Rightarrow \log y=\sin ^2 x \Rightarrow y=e^{\sin ^2 x}
$

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