Homogeneous Differential Equation

A curve passes through the point $\left(1,\frac{\pi }{6}\right).$ Let the slope of the curve at each point $\left(x,y\right)$ be $\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0.$ Then, the equation of the curve is

The substitution $y=z^{\alpha }$ transforms the differential equation $\left(x^2{y}^2-1\right)dy+2x{y}^{3}dx=0$ into a homogeneous differential equation for

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

The $x-$intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point $\left(1,1\right)$ is

A curve passes through the point $\left(1,\frac{\mathrm{\pi }}{4}\right)$ and its slope at any point is given by $\frac{y}{x}-{\cos }^2\left(\frac{y}{x}\right)$. Then, the curve has the equation

The curve for which the ratio of the length of the segment intercepted by any tangent on the $y-$axis to the length of the radius vector is constant $\left(k\right)$ is:

A curve passes through $\left(2,1\right)$ and is such that the square of the ordinate is twice the rectangle contained by the abscissa and the intercept of the normal, then the equation of curve is:

The solution of $\frac{dy}{dx}=\frac{-\cos x\left(3cosy-7\sin x-3\right)}{\sin y\left(3sinx-7\cos y+7\right)}$ is

The solution of $\frac{dy}{dx}={\left(\frac{x+2y-3}{2x+y+3}\right)}^2 is$

The solution of $\frac{dy}{dx}=\frac{{\left(x-1\right)}^2+{\left(y-2\right)}^{2}ta{n}^{-1}\left(\frac{y-2}{x-1}\right)}{\left(xy-2x-y+2\right)ta{n}^{-1}\left(\frac{y-2}{x-1}\right)}$ is equal to

If $\left(y^3-2x^{2}y\right)dx+\left(2x{y}^2-x^3\right)dy=0$, then the value of $xy\sqrt{y^2-x^2}$ is

A function $y=f\left(x\right)$ satisfying the differential equation $\frac{dy}{dx}\sin x-y\cos x+\frac{{\sin }^{2}x}{x^2}=0$ is such that, $y\to 0$ as $x\to \infty$, then which of the following statements is/are correct?

Identify the statement(s) which is/are true?

The general solution of the differential equation, $x\left(\frac{dy}{dx}\right)=y·\log \left(\frac{y}{x}\right)$ is (where $C$ is an arbitrary constant)