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:
\(y=xcos^{-1}α+1\)
\(⇒y-\frac{1}{x}=cos^{-1}α\)
\(⇒cos(\frac{y-1}{x})=α\)