Question

For the differential equations, find a particular solution satisfying the given condition:$cos(\frac{dy}{dx})=α(α∈R);y=1$ when x=0

Answer 1

$cos(\frac{dy}{dx})=α$
$⇒\frac{dy}{dx}=cos^{-1}α$
$⇒dy=cos^{-1}α\,\,dx$
Integrating both sides,we get:
$∫dy=cos^{-1}α ∫dx$
$⇒y=cos^{-1}α.x+C$
$⇒y=xcos^{-1}α+C...(1)$
Now,y=1,when x=0.
$⇒1=0.cos^{-1}α+C$
⇒C=1
Substituting C=1 in equation(1),we get:
$y=xcos^{-1}α+1$
$⇒y-\frac{1}{x}=cos^{-1}α$
$⇒cos(\frac{y-1}{x})=α$