Question

For the differential equation in given question find a particular solution satisfying the given condition.​​

$cos\left(\frac{dy}{dx}\right)=a\left(aϵR\right),y=2whenx=0.$

Answer 1

Given,
$cos\left(\frac{dy}{dx}\right)=a\Rightarrow \frac{dy}{dx}=co{s}^{-1}a$
On separating the variables, we get $dy=\left(co{s}^{-1}a\right)dx$
On integrating both sides, we get $\int dy=co{s}^{-1}a\int dx\Rightarrow y=co{s}^{-1}a\left(x\right)+C...\left(i\right)$
On putting y=2 and x=0, we get $2=\left(co{s}^{-1}\right(a\left)\right)\left(0\right)+C\Rightarrow C=2$
On substituting C=2 in Eq. (i), we get
$y=xco{s}^{-1}a+2\Rightarrow \frac{y-2}{x}co{s}^{-1}a\Rightarrow cos\left(\frac{y-2}{x}\right)=a$
which is the required particular solution.