Animated Solution for Mathematics - Differential Equations: Let $x=x(y)$ be the solution of the differential equation $y=(x-y\frac{dx}{dy})\sin (\frac{x}{y})$ $y>0$ and $x(1)=\frac{\pi }{2}$ Then $\cos (x(2))$ is equal to :

The given differential equation is: We are provided with the initial condition $x(1)=\frac{\pi }{2}$ and our goal is to determine the value of $\cos (x(2))$.

Observe the argument inside the sine function, $\frac{x}{y}$. This suggests that the equation can be simplified by expressing it in terms of this ratio. Dividing both sides by $y$ (assuming $y>0$), we obtain:

Recall the quotient rule for the derivative of $\frac{x}{y}$ with respect to $y$:

This allows us to rewrite the bracketed term as $-y\frac{d}{dy}(\frac{x}{y})$.

Substituting this expression back into our equation yields:

Rearranging the terms to separate the variables, we get:

Integrating both sides, we find: Multiplying by $-1$, the general solution is:

Using the condition $x(1)=\frac{\pi }{2}$, we substitute $y=1$ and $x=\frac{\pi }{2}$ into the general solution: Thus, $C=0$, and our particular solution is $\cos (\frac{x}{y})=\ln y$.

To find $\cos (x)$ when $y=2$, we first note that $\cos (\frac{x}{2})=\ln 2$. We apply the double angle identity $\cos (x)=2{\cos }^2(\frac{x}{2})-1$:

The final result is: