Method of undetermined coefficients is a little fuzzy Can someone please remind me how to solve the following ODE
$$-4\frac{dv}{dx}-4\pi^2v(x)=\frac{1}{10}\sin(\pi x)$$
 A: Find the solution of the homogenous equation (i.e. without right hand side). Hint: exponent.
Find any particular solution. Hint: try sinuses.
Take the sum.
A: 1) Find the general solution of the associated homogenous equation (that is, the equation we get when we replace $\frac{1}{10}\sin(\pi x)$ by $0$.) 
You may be able to write down the general solution of the homogeneous equation immediately. It may help  to rewrite the equation as $\frac{dv}{dx}=-\pi^2 v$. What functions have derivative equal to a constant times themselves?
2) Find a particular solution of the original equation. (We address that later.)
3) Add.
A particular solution: Look for a solution $v(x)$ of the shape 
$$v(x)=A\cos(\pi x)+B\sin(\pi x).$$
 We want to find $A$ and $B$ that work.
Note that for this choice of $v$, we have
$$\frac{dv}{dx}=-\pi A \sin(\pi x)+\pi B \cos(\pi x).$$
Substitute $\frac{dv}{dx}$, $v$ in the left-hand side of our equation. The result should be identically equal to $\frac{1}{10}\sin(\pi x)$.
That means the coefficient of $\cos(\pi x)$ should be $0$, and the coefficient of $\sin(\pi x)$ should be $1$. 
That will give you two linear equations in the two unkowns $A$ and $B$. solve.  
