Solve the equation below using a two$-$parameter trial solution by

$($a$)$ the point collocation method setting the domain residual $R_d=0$ at $x=\frac{1}{3}$ and $x=\frac{2}{3}$ and

$($b$)$ the Galerkin weighted residual method

$($c$)$ Find an exact solution using separation of variables $($learned in your differential equations

class$).$

$($d$)$ Plot the solutions for $y$ obtained in $($a$),($b$)$ and $($c$)$ versus $x$ to determine which method

$($either point collocation or the Galerkin weighted residual method$)$ is best.

$($c$)$ Plot the domain residuals $R_d$ vs$.x$ for the methods in $($a$)$ and $($b$)$ and discuss which method

does best at minimizing the residual.

$\frac{dy}{dx}+y=0,0x1$

$y(0)=1$