Solve a second order DEQ using Euler's method in MATLAB I need to solve the equation below with Euler's method: 

$$y''+ \pi ye^{x/3}(2y'  \sin(\pi x)+\pi y\cos (\pi x)) = \frac{y}{9}$$ for the initial conditions $y(0)=1$, $y'(0)=-1/3$

So I know I need to turn the problem into a system of two first order differential equations. 
Therefore $u_1=y'$ and $u_2=y''$ I can now write the system as: 
$$u_1=y' \\ u_2=\dfrac{y}{9}-\pi y e^{x/3}(2u_1  \sin(\pi x)-\pi y\cos (\pi x))$$
How do I proceed from here? 
 A: Letting $u=y'$ is the right idea. This gives you the pair of equations
\begin{align*}
u' &= \frac{y}{9} - \pi ye^{x/3}(2u\sin(\pi x) + \pi y\cos(\pi x))\\
y' &= u
\end{align*}
which is a standard initial value problem. Notice that there are no $x$ derivatives, so you can integrate each value of $x$ separately. So fix some value for $x$ and discretize the system using your favorite method -- I am a big fan of Velocity Verlet:
\begin{align*}
y_{n+1} &= y_n + hu_n\\
u_{n+1} &= u_n + h \left(\frac{y_{n+1}}{9}-\pi y_{n+1}e^{x/3}(2u_n\sin(\pi x)+\pi y_{n+1}\cos(\pi x))\right),
\end{align*}
where $h$ is the time step; you could also use e.g. forward or backward Euler (you say you want "Euler's method" but there are at least three!). You know $u_0$ and $v_0$ from the initial conditions, so just iteratively apply the above rules to trace out the trajectory through time.
EDIT: Forward Euler would be
\begin{align*}
u_{n+1} &= u_n + h \left(\frac{y_{n}}{9}-\pi y_{n}e^{x/3}(2u_n\sin(\pi x)+\pi y_{n}\cos(\pi x))\right),\\
y_{n+1} &= y_n + hu_{n+1}
\end{align*}
Notice that I strongly recommend against using Explicit Euler in pretty much any circumstance.
