Find the general solution of the following differential equation: $x^{2}\frac{dy}{dx}=x^{2}+xy+y^{2}$.

The order of the differential equation \(\frac{d^{4}y}{dx^{4}}-sin(\frac{d^{2}y}{dx^{2}})=5\) is:

Solve the differential equation given by:$$x \, dy - y \, dx - \sqrt{x^{2} + y^{2}} \, dx = 0$$

The differential equation \(\frac{dy}{dx}=F(x,y)\) will not be a homogeneous differential equation, if \(F(x,y)\) is :

(i) Write the order and degree of the given differential equation. (ii) Substituting $V = \pi r^3$ and $S = 2\pi r^2$, we get the differential equation $\frac{dr}{dt} = \frac{2}{3}k$. Solve it, given that $r(0) = 5$ mm. (iii) (a) If it is given that $r = 3$ mm when $t = 1$ hour, find the value of k. Hence, find t for $r = 0$ mm. OR (iii) (b) If it is given that $r = 1$ mm when $t = 1$ hour, find the value of k. Hence, find t for $r = 0$ mm.