Question:
If $x \phi(x)=\int_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x>-2$, and $\phi(0)=4$ then $\phi(2)$ is
Solution:
$x \phi(x)=\int_{5}^{x} 3 t^{2}-2 \phi^{\prime}(t) d t$
$x \phi(x)=x^{3}-125-2[\phi(x)-\phi(5)]$
$x \phi(x)=x^{3}-125-2 \phi(x)-2 \phi(5)$
$\phi(0)=4 \Rightarrow \phi(5)=-\frac{133}{2}$
$\phi(x)=\frac{x^{3}+8}{x+2}$
$\phi(2)=4$