1.

xcos (y/x) dy/dx=ycos (y/x) - x

Answer»

x cos \(\frac{y}{x}\) \(\frac{dy}{dx}\) = y cos \(\frac{y}{x}\) - x

⇒ \(\frac{dy}{dx}\) = \(\frac{y\,cos\,\frac{y}{x}-x}{x\,cos\,\frac{y}{x}}\) = \(\frac{\frac{y}{x}\,cos\,\frac{y}{x}-1}{x\,cos\,\frac{y}{x}}\) 

Take y = vu

\(\frac{dy}{dx}\) = v + x\(\frac{dv}{dx}\)

∴ v + x\(\frac{dv}{dx}\) = \(\frac{v\,cosv - 1 }{cosv}\) 

⇒  x\(\frac{dv}{dx}\) = \(\frac{v\,cosv - 1 }{cosv}\) - v = \(\frac{v\,cosv - 1 -v\,cosv}{cosv}\)

⇒  x\(\frac{dv}{dx}\) = \(\frac{-1}{cosv}\)

⇒ cosvdv = \(\frac{-1}{x}\)dx

⇒ - sin v = \(\frac{1}{x^2}\) + c

(By integrating both sides)

⇒ - sin \(\frac{y}{x}\) = \(\frac{-1}{x^2}\) + c .....(1)

(By putting  v = \(\frac{y}{x}\))

Equation (1) represent solution of given differential equation.


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