If $x \phi(x) = \int_{5}^{x} (3t^{2} - 2 \phi'(t)) dt$,$x > -2$,and $\phi(0) = 4$,then $\phi(2)$ is .... .

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and $\lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1$ for each $x>0$. Then,for all $x>0, f(x)$ is equal to

Let $x=x(y)$ be the solution of the differential equation $2 y e^{x / y^{2}} d x+\left(y^{2}-4 x e^{x / y^{2}}\right) d y=0$ such that $x(1)=0$. Then,$x(e)$ is equal to

Let $y=y(x)$ be the solution curve of the differential equation $(1+x^{2})dy+(y-\tan^{-1}x)dx=0$, with $y(0)=1$. Then the value of $y(1)$ is:

The solution of the differential equation $\frac{dy}{dx} + y = \cos x$ is

An integrating factor of the equation $(1+y+x^2 y) dx+(x+x^3) dy=0$ is