The solution of $\frac{dy}{dx} = \left( \frac{y}{x} \right)^{1/3}$ is

Let $f$ be a non-negative function defined on the interval $[0,1]$. If $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ for $0 \leq x \leq 1$ and $f(0)=0$,then:

If the solution of $\frac{dy}{dx} = \frac{y^3 \cos \sqrt{x}}{\sqrt{x} e^{1/y^2}}$ with $y(0) = 1$ is $\frac{1}{y^2} = \log_e(f(x))$,then $f(x) =$

If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$ with $y(0)=3$ is $\alpha x+\beta y+3 \log_e|2x+3y-\gamma|=6$,then $\alpha+2\beta+3\gamma$ is equal to

General solution of the differential equation $\frac{dy}{dx} = \frac{x+y+1}{x+y-1}$ is given by

The solution of $\frac{dy}{dx} = 2^{y - x}$ is