The particular solution of the differential equation $\cos \left(\frac{\mathrm{dy}}{\mathrm{d} x}\right)=a$,…
The particular solution of the differential equation $\cos \left(\frac{\mathrm{dy}}{\mathrm{d} x}\right)=a$, under the conditions $\mathrm{a} \epsilon \mathrm{R}$ and $y(0)=2$ is
$\cos \left(\frac{x-2}{y-2}\right)=a$
$\cos ^{-1}\left(\frac{y-2}{x}\right)=a$
$\cos \left(\frac{y-2}{x}\right)=a$
$\cos \left(\frac{x-2}{y+2}\right)=a$
Solution
$\begin{aligned} & \cos \left(\frac{d y}{d x}\right)=a \Rightarrow \frac{d y}{d x}=\cos ^{-1} a \\ & \int d y=\int \cos ^{-1} a d x \\ \therefore & y=x \cos ^{-1} a+c ....(1)\\ & \text { We have } x=0, y=2 \\ \therefore \quad & 2=0+c \Rightarrow c=2 \\ \therefore \quad & y=x \cos ^{-1} a+2 \\ \therefore & y-2=x \cos ^{-1} a \Rightarrow \frac{y-2}{x}=\cos ^{-1} a \Rightarrow \cos \left(\frac{y-2}{x}\right)=a \end{aligned}$