Proceed as in Example $6$ to solve the given initial$-$value problem.

$\frac{dy}{dx}+P(x)y=0,y(0)=5,$ where

$P(x)=\{\begin{array}{cc}1,& 0x2\end{array}$

$4,x>2$

On the interval $0,2,$ find the coefficient function $P(x)$ when the given differential equation is written in the standard form $\frac{dy}{dx}+P(x)y=f(x).$

$P(x)=$

Find the integrating factor for the differential equation.

$e^{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}P(x)dx}=$

Find an explicit solution to the initial$-$value problem on the interval $0,2.$

$y=,on0,2$

Find $y(2).$

$y(2)=$

Find an explicit solution to the initial$-$value problem on $[0,).$

$y=\{\begin{array}{cc},& 0x2\\ ,& x>2\end{array}$