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  4. The particular solution of the equation, (d y/d x)= cos (x+y), at (0, π / 2) is

Q. The particular solution of the equation, $\frac{d y}{d x}=\cos (x+y)$, at $(0, \pi / 2)$ is

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Solution:

Put $x+y=v$.
Then $1+\frac{d y}{d x}=\frac{d v}{d x}$.
$\therefore \frac{d y}{d x}=\cos (x+y)$
$ \Rightarrow \frac{d v}{d x}-1=\cos v $
$\Rightarrow \frac{d v}{d x}=2 \cdot \cos ^{2}\left(\frac{v}{2}\right)$
$\Rightarrow \frac{1}{2} \sec ^{2}\left(\frac{v}{2}\right) \cdot d v=d x$.
By integrating we get, $\tan \frac{v}{2}=x+c$
$\Rightarrow v =2 \tan ^{-1}( x + c ) $
$\Rightarrow x + y =2 \tan ^{-1}( x + c )$.
When $x=0$ and $y=\pi / 2$ we get $c=1$.
Thus the particular solution is $y =2 \tan ^{-1}( x +1)- x$.

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