If x(x)=_5^x(3t^2-2'(t)),dt, x-2, and (0)=4, then (2) is Answer: 4

Question

If $x\phi(x)=\displaystyle\int_5^x(3t^2-2\phi'(t))\,dt$, $x>-2$, and $\phi(0)=4$, then $\phi(2)$ is

Answer: 4

Accepted Answer

$$x\phi(x)=\int_5^x(3t^2-2\phi'(t))dt=\left[t^3-2\phi(t)\right]_5^x=(x^3-125)-2[\phi(x)-\phi(5)]$$

At $x=0$: $0=-125-2[4-\phi(5)]\Rightarrow\phi(5)=\dfrac{133}2$

At $x=2$: $2\phi(2)=(8-125)-2\left[\phi(2)-\dfrac{133}2\right]\Rightarrow 4\phi(2)=16\Rightarrow\phi(2)=4$

Solutions · 1