If $x \sin \left(\frac{y}{x}\right) dy = \left[y \sin \left(\frac{y}{x}\right) - x\right] dx$, $x > 0$ and $y(1) = \frac{\pi}{2}$, then the value of $\cos \left(\frac{y}{x}\right)$ is

If $\sin \left(\frac{y}{x}\right)=\log |x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi}{3}$,then $\alpha^2$ is equal to

The real value of $m$ for which the substitution $y = u^m$ will transform the differential equation $2x^4y \frac{dy}{dx} + y^4 = 4x^6$ into a homogeneous equation is:

The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{\phi(y/x)}{\phi'(y/x)}$ is

The substitution required to reduce the differential equation $t^2 dx + (x^2 - tx + t^2) dt = 0$ to a differential equation which can be solved by the variables separable method is

Show that the differential equation $\left(1+e^{\frac{x}{y}}\right) dx+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) dy=0$ is a homogeneous equation and find its solution.