Differential Eq...

For $$x\epsilon R, x\neq 0$$, if $$y(x)$$ is a differentiable function such that $$x\int_{1}^{x}y (t) dt = (x + 1) \int_{1}^{x} t y (t) dt$$, then $$y (x)$$ equals:
(Where C is a constant)

Let $$y = y(x)$$ be a solution of the differential equation, $$\sqrt{1-x^2}\dfrac{dy}{dx} + \sqrt{1-y^2} = 0, |x| < 1$$.
If $$y\left(\dfrac{1}{2}\right) = \dfrac{\sqrt{3}}{2}$$, then $$y \left(\dfrac{-1}{\sqrt{2}}\right)$$ is equal to:

The differential equation $$\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}=\frac{\sqrt{1-\mathrm{y}^{2}}}{\mathrm{y}}$$ determines a family of circles with:

Check whether the function is homogenous or not. If yes then find the degree of the function
$$g(x)=x^2-8x^3$$.

Solution of differential equation $$\displaystyle \frac{dy}{dx} = sin  x  + 2x$$, is

The order of the differential equation
$$2x^2\dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + y = 0$$ is

The solution of $$\dfrac{dy}{dx}=e^{logx}$$ is:

The solution of $$\dfrac{dy}{dx}=\dfrac{2x}{3y^{2}}$$ is:

The solution of $$\dfrac{dy}{dx}-\dfrac{2xy}{1+x^{2}}=0$$ is

The solution of $$x^{2} \cfrac{dy}{dx}=2$$ is