First slide

Methods of solving first order first degree differential equations

Question

$The solution of differential equation e^{- x} (y + 1) d y + (\cos^{2} x - \sin 2 x) y d x = 0, if y (0) = 1, is$

Moderate

A

$(y + 1) + e^{x} \cos^{2} x = 2$
B

$y + \log y = e^{x} \cos^{2} x$
C

$\log (y + 1) + e^{x} \cos^{2} x = 1$
D

$y + \log y + e^{x} \cos^{2} x = 2$

Solution

$Given that e^{- x} (y + 1) d y + (\cos^{2} x - \sin 2 x) y d x = 0$
$\begin{array}{l} \Rightarrow \frac{y + 1}{y} d y = - e^{x} (\cos^{2} x - \sin 2 x) d x \\ Integrating on both sides \\ \int (1 + \frac{1}{y}) d y = - \int e^{x} [\cos^{2} x - \sin 2 x] d x \\ \Rightarrow y + \log y = - e^{x} \cos^{2} x + c \dots \dots . . 1) (∵ \int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x)) \end{array}$
$\begin{array}{l} As y (0) = 1 \\ 1 + \log (1) = - e^{0} \cos^{2} 0 + c \\ \Rightarrow y + \log y + e^{x} \cos^{2} x = 2 \end{array}$

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